Unfortunately, the previous guide for this achievement has disappeared, which is a shame because it was really good. This will be my own attempt to help guide people through this process. Here's the long and not-short of it.
As the description suggests, this achievement is for researching but not necessarily owning/purchasing a tier-X vehicle of all five classes: light, medium, heavy, tank destroyer, and artillery. [See Notes at the end of this solution.] This will be a long slog made no easier by the fact that not all tanks are created equal. But there are two ways of going about this. First, the shorter way.
As Cubivore pointed out in the comments, the Chinese medium line splits off at tier-VII into both their medium and heavy lines. This cuts down on the amount of time needed to grind two separate lines, although tiers-I through -VI are all relatively quick compared to high tier grinding. However, all four Chinese lines (light, medium, heavy, and tank destroyer) have some overlap with the others. This means that researching a module on one line that the other lines also share will unlock that module on the other lines. This means you could potentially get two, three, or even four times the benefit from a single research package, so long as that radio, gun, hull, whatever is shared across lines. This also has the benefit of allowing you to use the same crew, which spares you the trouble of having to train up a crew for multiple nationalities.
On the other hand, you could decide to go the long way and use tanks from different nations. The "best tanks" for each particular class aren't from the same nation. If you choose to forego sticking exclusively with the Chinese tanks, then you'll have to diversify and go with different nationalities for each tank. Here are, in my opinion, best tank lines by class and nationality.
For this one, you have two obvious "best" choices: Soviet or German tanks.
Soviets - Generally speaking, the Soviet tanks are better because the sloped hulls on the IS-7 line make it very easy to to bounce shots, and they have fairly good guns. This increases your survivability and your ability to gain experience. The IS-4 line trades the superior armor for better guns, if that's your thing. Make sure to protect your lower plate and sides.
Germans - These guys are more difficult to play than the Soviet tanks because they rely more on "advanced" tactics like side-scraping and hull-down positioning. Like the Soviets, their front underplating is their greatest weakness, but if you can master side-scraping (i.e. angling your armor so that the enemy shells scrape off your side) then you're good. Much stronger guns than the Soviet lines, on average, but longer reload times. The E-100 line is the way to go if you want a stronger gun, and the Maus line is best for those who want good armor and lots of hit points. All three German heavy tank lines are what are known as "super-heavies," and they're worthy of the name.
Notable mentions: French heavies are lacking in armor but have fantastic guns. If you're comfortable using auto-loaders, consider going with the French, but don't try to rely on your armor. You'll bounce some shots, but that will just be because the RNG gods are feeling merciful. The Japanese also have super-heavy tanks starting from tier-V, and their hit points rival the German E-100 and Maus lines. Their guns fire very slowly, though, and while you'll be regularly one-shotting people in the mid-tier game, at higher tiers you'll find that their guns just don't keep up with the competition.
This one is a lot more complicated than the others because you have so many different kinds of mediums, from ones that specialize in flanking to others that are closer to being heavies to others that are just light tanks with slightly more hit points.
British and Soviet - These guys are closest to being heavy tanks, so use them if you like that play style, the Soviets more so than the British. The Soviet Obj. 430 line gives you the most survivability in my experience, but the British Centurion AX line is no slouch either. The British fire faster but deal less damage than the Soviets.
American and Japanese - These guys are flankers. Their armor is decent, though not as good as the first pair. The American mediums surpass the Japanese when it comes to armor, though the Japanese have slightly better damage and accuracy overall. The better guns on the Japanese might not make up for their thinner armor and worse hull profiles that make them more susceptible to point-blank annihilation, however.
Honorable mentions: The Czechoslovakian and French tanks both favor auto-loading tanks, and the French are some of the fastest mediums out there. Again, if you're comfortable with auto-loaders and shooting on the run, consider giving them a try. The French play more like light tanks than any of the others.
For me, this class is complicated. The fixed hulls mean you'll want to stay farther back, and some of them have really bad armor despite being so slow and unwieldy. But some are better than others.
British or German - The British FV215b (183) tank line probably has the best armor out of all the tank destroyers, although they're very slow and weak at the sides and the commander's cupola (the hatch). The top one in the line, the FV-blah-blah-blah, has been nicknamed the Deathstar because of its destructive firepower, but if you're only going for the research, you won't get to enjoy the Anti-Alderaan Cannon. The German Grille 15/WT auf E 100 lines boast more firepower than the Jg. Pz. E100 line does, but it comes at the cost of the latter's superior armor. The WT auf E 100 (aka the Waffle) is considered one of the most overpowered vehicles in the game, but like the Deathstar, you won't get to enjoy it much if you're just going for the research.
Americans - The T110E5 line overcomes what I view to be the biggest weakness of the tank destroyers: the fixed hull. The German Waffle line also have rotating turrets from tier-VIII onward, but the T110E5 line sports a non-fixed hull as soon as tier-IV. Be aware that these guys are not as durable or as powerful as the previous two, however. Play with caution.
Honorable mentions: The French Foch and Foch 155 have very good frontal armor, and overall the line has very decent guns. Their lower hit points are their major drawback, however. The Swedish tank destroyers have what is known as "siege mode," which gives them better accuracy and aim time at the expense of speed and maneuverability. Siege mode is activated/deactivated by pressing left on the d-pad, and it takes a few moments to switch from one to the other.
American - While the Soviets and Germans compete for the title of best heavy, the Americans are hands down the best artillery in the game. There is no competition when it comes to which is the best of the best in the artillery line, though if you want to settle for "good enough," the Germans get that title. Incredible damage, very good accuracy, decent loading and aiming times, excellent speed, and superior range of motion all add up to a devastating artillery class.
Notable mentions: The French line is an anomaly amongst the artillery lines. Their damage is fairly low, but the reason I'm mentioning them is because they have the best reload times and accuracy. This alleviates the problem of having to wait 45 seconds between shots only to miss because your target moved at the last second. Also, the relatively lower reload time means you can "perma-track" enemies so that your allies can go to town on them. Consider these guys if you're feeling adventurous.
As the newest class to go up to tier-X, it's harder to get a feel for which is "the best" of the light tanks.
French and Germans - These two are fastest of them all, and the French auto-loaders are a beast to be feared. This is the way to go if you prefer active scouting, i.e. constantly moving to expose the enemy while dodging shots. The long reload between ammo clips for the French means you'll have somewhat lengthy bits of time when you simply can't fire, so move around spotting enemies during that time.
Americans - In my experience, the Americans are the best at damage per shot, and they have decent sight range. Probably better for passive spotting and sniping, though the smaller maps on console make passive spotting (i.e. trying to stay in one place to provide sight to your allies) less viable than on PC, because it's easier for the enemy to get close to you and wipe you out. The American lights often use the same guns as their medium counterparts.
Chinese and Soviets - These guys go with their typical "better armor, worse mobility/spotting" trade off. They can be devastating, and their armor is more likely to bounce enemy shots, but don't try to rely on that too much. They're still light tanks, after all. Between the two, the Chinese are better at shooting on the go, so doing circles-of-death around slower enemies is a possibility. Definitely the worst out of all of them when it comes to scouting, however.
And so there it is. Perhaps not as good as the previous solution, but hopefully it's enough to at least guide people through this long, long, long achievement. To wrap it up, here are just some general miscellaneous tips.
[Important! When I researched the fifth and final class, the achievement did not unlock. I played another game in a separate tank, but nothing. Then I bought the two tank classes I had researched but not purchased and played a game in each, but still the achievement did not unlock. Finally, I quit the game and restarted, and the achievement popped when I signed back into the game. Didn't even have to go back to the garage. If the achievement does not unlock right away, try quitting the game completely and then logging in, or some combination of what I did above.]
1.) Mentor helps with crew training but not battle XP. This means that while it will help you get a better crew faster, it won't directly help you research the tanks. Still, a better crew makes for easier matches.
2.) Premium ammo doesn't do more damage than your normal ammo, but it does have better penetration and generally travels faster. Since the majority of your XP comes from how much damage you do, maybe consider having a few rounds prepared for when you're facing super-heavies or other difficult foes. Select the ammo and press Y under the supplies tab to switch between gold and silver for resupply. You still have to make sure to aim for weak points, though. Premium ammo isn't omnipotent.
3.) You get full XP for spotting and damaging your own targets but only half XP if someone assists you, either by spotting the target for you or by immobilizing it. By the same token, you also get half XP if you're the spotter or if you immobilize the target. Consider being a spotter or a tracker if you're having trouble dealing damage, especially when you're low tier in a match or have the stock package.
4.) War Stories takes longer than multiplayer, but it can be a good way to grind through unpleasant tanks or stock packages. This is definitely not true if you're playing artillery, though, as your allies (if you have any) won't move forward unless you do. WarGaming is also constantly updating War Stories, meaning it gets more difficult all the time.
5.) Know when you can go it alone and when you should stay with the pack. Certain tanks, like the Japanese super-heavies and the fixed hull tank destroyers, should never be alone, lest the enemy get too close and chip their health away.
6.) There is no number 6.
7.) Platooning can make things easier thanks to your ability to communicate and coordinate. If you get a good group together, you might even have fun doing it.
8.) WarGaming regularly gives away premium time. Look for ops to give yourself a free boost, or consider purchasing premium time at the store if you feel like supporting the game.
9.) Just Google it. There are plenty of almost-professional World of Tanks players out there with their own YouTube channels that can help point you in the right direction for each and every tank in the game. If you're having trouble with a particular tank, look through the internet for tips on how to play it. Be advised that a lot of them are on PC, so there may be some discrepancies. Still, the differences shouldn't be so large that the YouTubers can't help you.
10.) Keep at it, but know when to take breaks. Don't let the game burn you out. Play as long as it's fun/tolerable, but have some other games ready for when you just aren't feeling it.
Final Side Note: I'm anticipating that WarGaming will eventually add an achievement that requires researching tier-X vehicles from multiple nationalities, so I'm stacking this current achievement with that potential future achievement. Although you can cut down on a lot of time by researching tanks within the same nationality, I'm taking steps just in case WarGaming does create the achievement I mentioned. It's something to consider.