You must win the 'Largest Army' award 10 times in matches. Like the longest road award you will need to win an award in the game to unlock this, although for this you must use three soldier development cards during one match to get the award. When you set up a match make sure that the AI you choose do not have a soldier icon next to their name (it is a red/brown man) which indicates they will use soldier cards more often, avoiding these players mean you have a greater chance of winning the award.

Collect wool, grain and ore resource cards and keep buying development cards until you have three soldiers. There are 14 soldier cards out of the 25 development cards available so you will find they are easier to obtain than the other cards. Then all you need to do is play three cards during three of your turns and you win the award, make sure to keep playing soldier cards if any of the AI play one.

You can check how many a player has used by looking at the number of chevrons next to their picture, one chevron is one soldier card played.