DISCLAIMER: You have to complete story mode to unlock all of the maps for play in custom games.
Start out with the 2 player maps, and work your way up. If you play a 2 player map that also has a 3 player version of the same map, it counts it as playing the map on all versions. If you play the Classic risk map on 3 player, it also counts for the 4 and 5 player versions.
I found playing the lowest possible players on a map was easiest, as it didn't take as long since more players= taking more turns, as well as less territories to start the game with, and just less people to worry about having to kill.
There is a star next to a map that when lit up, means you beat the map.
EDIT: I'm going to start posting strategies for each map. EDIT 2: Only got around to Classic Risk, but that was the hardest one to beat anyways.
All maps are played in Factions mode, and you don't play Classic Risk in Classic Risk mode.
I'd suggest playing on the 3 player variant, as this map will take quite a while, and can be pretty hard.
Pick manual setup. This is crutial to the strategy. You want to pick Madagascar as your first territory. If the computer player picks it, restart. This city will play a key point in eliminating one of the other players. Don't take any other cities in Africa during the city placement. Next you'll want to get as many cities in Australia as you can. If you can get all 4, that's even better. Lastly, place your remaining starting positions in a line connecting Australia to Russia, as close to the sea jump to the USA as you can get (or take it if the computer hasn't). Japan is a city you want to start out with, but can do just fine without it. If you still have remaining troops, bridge Australia to Arfica, or just place them wherever, they won't be of any use to you if they aren't connected to you. Place your Capital on whatever city in Australia you beefed up
Now that city placement is done, it's time for troop placement. Place 5-6 guys on Madagascar, then the rest in Australia on one city (if the computer has any cities in Australia, only place 4 on Madagascar, and put the rest on a city next to the one your opponent has). It would be smart to place them on the city that connects to Asia if/when you control Australia. DO NOT spread out your troops in Australia, that is possibly the worst thing you could do at the start of the game.
As soon as your turn rolls around, take the rest of Australia if you didn't start with it all. Place all but one troop on whatever city in Australia you beefed up, and place one on Madagascar. If you already started with all of Australia, place everything on Madagascar. If you have 10-12 troops on Madagascar (depending on what the troop numbers look like in Africa), you can probably take the whole continent in one attack. If you don't have 10-12, attack one city for a star, and end your turn, moving all the troops back to Madagascar.
As each turn starts, split your draft 50/50 between your beefed up city in Australia, and Madagascar, until you take Africa, or have the two cities connected to Madagascar (would be pretty pointless to put troops on a city you can't attack from), then when the enemy troop numbers look favorable, attack from Madagascar, and take the whole continent. Once you take it, if you have enough troops to do so, split up the troops as evenly as possible between the city that connects to Europe, the city that connects to South America, and the city that connects to Asia (I think it's Egypt). If not, put them on the Asian bordered city.
If you hold two continents, your next troop placement should be strengthening your borders in Africa, until you have 5 on each bordered city to be safe. Once that has been achieved, start sweeping through Asia. Start placing all of your troop positions on your Asia bordered city in Africa, and work your way to the sea jump to USA. You'll want to leave 6-7 troops here at all times when possible. If you can get all of Asia, it should be a cakewalk to clean up the remaining cities with 3 continent bonuses rolling in on top of normal draft numbers.
If there is a badge thing for taking X cities in one turn, and you haven't taken Africa yet, wait until you have 15 troops on Madagascar, and bowl through Africa, then take one or two cities in Europe or South America to keep the opponents from possibly getting a bonus in that continent (Europe gives 5, which is pretty decent compared to you getting 5 with two continents)
Ignore cards that require you to take four buildings (little office looking buildings on the terrain) in one turn. Usually youd have to have either a huge army placement (in which case you'd be better off taking over a continent or two), or you'll already own too many cities with buildings on them.
For taking over enemy capitals, all you have to do is hope they place them in Australia, Africa, or Asia. If not, just ignore that card until you can get close enough to one without possibly losing a ton of your cities trying to take it (if the computer beefed up their capital)
Always keep at least 10 troops on your Asia-Australia bordered city. It can be a pain if you lose Australia and have to waste a whole turn trying to take it back.
Try to avoid cashing in troops until you have 10 stars. If you absolutely have to (if you are close to dead, or someone is camping in Australia with a big troop placement), focus on taking back cities, and not attacking other continents. Also, you'll want to take out the bigger troop placements over retaking cities with only 1 troop on them, otherwise they can just demolish you during the next draft, and you'll lose at least 1 guy due to leaving one behind to hold the city for each city you take.