Risk: Factions

The easiest way to get this achievement is to play a 5 player game of world conquest. You need to control the whole map to win the game and as such you'll get five capitals in the process.

My personal strategy for winning easily is to play on the 5-player map Limbo on manual and place my troops all along the left side and in Black Fort to capture Von's Temple. Then after gaining control of the left islands, it's really easy to push into the right and win, missile or no missle.

Start a 5 player game and get all the capitals! These are the larger buildings with the light beams coming out of the roof.

This is easiest on the 5th and final mission of the campaign where you can use the temple to 'convert' the capitals on the far side of the map to your side and use might to take the rest.

Just avoid getting those final objectives (always take risk cards) until you get the 5th capital.

Ed: and as noted in the comments by dropK1CK ninJA - take the last capital by force for the achievement - using the temples for the last one didn't work.

Ok so here is my 4 controller strategy.

Start up a custom game. Select Risk Factions. Select command room, Random, normal game. Next select 5 players on the Limbo map. Have your 3 other controllers join the game and start. Place your capitol where the 5th person (AI) can't really get to it. Now place your 3 other controller's capitols next to each other, but somewhere where you are really close and the AI is somewhat far away. When it's your turn take over as many of the capitols as you can in one go. When it's your other controllers turns just draft the men to one territory far away, then skip thier turn. Once you have taken over the 3 capitols draft as many men as you can next to or really close to the AI capitol. It shouldn't take much to overtake his capitol.

Bam! Achievement Unlocked!

Hope this helps anyone struggling with this achievement as I did.