Ok, let's address the elephant in the room first. There is a method to complete 5-6 challenges repeatedly and "shortcut" the "The Champion" achievement. We will not be following that method here. In my opinion, you don't need it. Due to the risk of accidentally deleting your cloud save, as you go offline and back online multiple times, that guide recommends saving "The Champion" as the last achievement you complete. As a result, you will need to work significantly harder to make up for the Overcharge you would otherwise receive for completing the challenges normally, compounding to a longer time to unlock weapons and collectable maps. The collectables in turn can be swapped for Amps which make it easier to defeat several challenges. Never mind the time lost while repeatedly grinding through the first hours of the game over and over just to unlock the exact same challenges... It took me 3 hours to get my 50, and my days of fast twitch reflexes are sailing away. Some of these challenges are a little annoying but with a little persistence are very manageable. There was 1 transversal challenge that I had to repeat 10-15 times and most everything else took 1-3 attempts. Worst comes to worst, the challenges in the DLCs (which I do recommend you pick up) count towards the achievement and provide effective alternate options.
Now that each and every one of you agrees with me, allow us to proceed.
If you've been completing these as you go through the campaign, you can tell which ones, if any, are already gold. Select the mission from the map and on the left side is the medal you earned.
Buck National
There are 4 challenges from Buck, one for each episode. Episode 3 is the only achievement in the base game, with the other 3 achievements coming through Title Update 2. Your objective is to reach certain point totals while using different kill methods. Only in the first episode do you need to follow the kill pattern though. You are most likely to succeed by maintaining your style meter and kill combo. I recommend bringing (1) the "Grind The Lightning" Epic Amp to protect you while grinding and (2) your trusty Hairspray Bomb with the 2nd amendment amp mentioned previously. Almost certainly, you've reached level 5 with it by now, if not, no biggie. Upgraded and amped Acid Sprinklers, TNTeddys, and Turret Copters will do a lot of heavy lifting in the middle episodes as well.
Episode 1:
You follow a pretty short kill pattern here with 3 minutes on the clock. Bounce > Grind > Blower > Train and repeat. To reach 400,000 points you need to make it through 4 cycles. Start by bouncing on your obvious bounceables and getting your kills. While the game directs you to the train tracks to grind, instead finish your bounce kills by jumping up to the rail between the two water tanks. This sets you up for the most difficult part of the cycle, the Blowers. Launch your bombs down below to collect your grind kills, and Blowers will spawn out of the platforms you are now grinding on (in later rounds, 1 or 2 will spawn on the rooftop of the buildings flanking your bounceable surfaces). The Blowers also have yellow target indicators helpfully painted on them should you lose track of one or two. Dispose of them, and when the prompt to hit the train switch comes, you should hopefully find yourself right above it, still grinding that rail between the water tanks. Dive bomb the switch then get out of the way, because the Pain Train takes no prisoners. Make your way to the bounceable surfaces and repeat the cycle. Once you get 25 kills with the Train, you should have enough points (likely sooner, if you maintain your style), so either keep going crazy or just survive and let the timer run out. The timing might be tight if you waste too much time on the Blowers, but killing them near their spawn points should reduce that time loss. Finishing this round earns you
Episode 2:
There is another kill pattern here (Grind > Melee > Undergrind > Bounce > Blower > Electric barrels), but it is largely unnecessary to follow. Just like the first episode, maintaining your style with multiple movement types will reward you with more damage and points. Plenty of enemies spawn in to keep your kill combo going. 500,000 points in 3 minutes is quite easy, despite its initial appearance and comparing it to the first episode. Anecdotally, it seemed that more OD spawn in during the electric barrel sequence, but that may have just been faulty pattern recognition on my part. The barrels are quite easy to detonate accidentally making it difficult to remain in that sequence anyway. Your Hairspray Bomb should easily carry you to
Episode 3:
Kill pattern is again irrelevant (Melee > Grind > Spawner > Bounce > Undergrind > Mugger > Pyro Geyser). Wrap up the 5 melee kills quickly, then stay up on the wire that runs along the main street, though throw in some undergrinds and bounces to keep that style up. There are plenty of OD for your Hairspray Bomb to devastate. Only the Muggers gave me trouble, as they would jump onto the wire and mess up the combo and style counters. If you reach their turn, wipe them out quickly. Toss out some Acid Sprinklers and Turret Copters near the spawn and choke points at either end of your wire allowing you to focus on a tighter area. The QA team's high score of 534,080 will topple meriting you
Episode 4:
Now for the easiest episode of the bunch. As quickly as you can get 5 melee kills, 10 grind kills, and 5 bounce kills. Ignore the Winger that spawns in. Hopefully, you noticed that the OD horde is spawning from one point. Head over there (the large blue building). Keep bouncing on the two trucks immediately in front of the spawn point and spam your Hairspray Bomb. Watch out for the Winger's Overcharge beam that can be easily avoided. Once the timer hits zero, you unlock
Bomb Delivery Challenge
8 missions. Each begins with a cut-scene showing you the drop locations. Ignore the order it gives and determine a more linear approach. Usually, you want to create a hook-like path of hitting the closest first, the higher elevated of the 2 medium distance locations, then the furthest location, before hooking back to the other medium distance location. Use your transversal techniques and hit these as quickly as possible. Of course, replay the mission if you did not get a gold, but midway through this list you receive
Points Challenge
6 missions of gathering points using your transversal techniques. I recommend taking the first attempt to familiarize yourself with the level, finding all the points and mentally creating paths to grab them as efficiently as possible. You will need to collect every point to get gold. Any time left at the end adds bonus points to your total. Use your boost and air dash
frequently.
Smash Stuff
6 missions similar to the points challenge but instead of collecting items, you smash them. Again, take the first attempt to familiarize yourself with the target locations (red dots on your mini-map) and determine efficient paths. For gold, you need to destroy each and every item. Explosive barrels, when available, and dive bombing offer a larger damage radius, but you will largely grind between or along targets. Frequently, smashing one item in a line will eliminate the entire row. Air dash and boost whenever possible.
Weapon Challenge
8 missions where you test out interesting weapons. Your old OP Excalamune, Murderang, and TNTeddy should come easily as they should be very familiar by now, as should the replay of the pigeon roasting mission (use the same strategy). You also meet, if you haven't purchased them already, the Charge Beam, the One Hand Dragon, The Dude, and The Shocker. The Charge Beam needs to warm up briefly before letting a steady laser out. One Hand Dragon is basically a smaller Hairspray Bomb. The Dude abides by slinging unstoppable bowling balls, so retreat to form the OD horde into a nice line as they chase you and unleash. The Shocker is a medium range focus weapon that halts enemy ingress by shocking them. Hop up on the rail inside the cooling tower and take out OD as they approach.
If you are following this order, midway through these challenges you should receive
Second PlaceComplete 30 challenges with at least a silver rating.

Glider Challenge
4 missions: #1 & #3 you target floating mines. Keep spamming your missiles
and turning for new targets. Watch your power meter and hit an air spout to boost it, if needed. #2 & #4 are similar to the transversal challenges. It appears as a time trial or race but hitting each ring in turn is enough to get gold. If you miss a ring, just crash to restart the attempt. It is not worth circling around. Be careful when navigating near buildings.
Night Defense
The same four Night Defense missions from the campaign but with the more powerful weapons you've collected.
Lay your traps and by now you should receive
and
Flung to SafetyKill 100 enemies by flinging them into danger using the Springboard Trap.

Since there is only one vat, the first mission in Old Factory is probably the easiest to save all your Overcharge and earn
PerfectionComplete an optional Night Defense without losing any of your Overcharge.

Transversal Challenge
14 missions where the goal is to follow the route, hitting all the rings as quickly as possible. I had the most trouble with a couple of these towards the end of the list, but if you've completed all of the previous challenges, you can drop 4 of these. Use all of your transversal techniques, including air dashing, boosting, and water running. Don't forget to round the corner while wall running using 
When you get your 10 golds, enjoy
If you had trouble with any mission types, MissFuchsiaFire's guide, listed under the achievement link, provides several helpful videos. If you own them, the DLCs contain several challenges which count towards this achievement. If need be, you can use the offline technique, but know that I'll know and be disappointed.
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