Command Points are used to purchase new Heroes, which are vital for building a strong team of Avengers and fighting your way through this game.
How to get Command Points:
If you are struggling to earn enough Command Points to spend in Avengers Alliance, you can always spend Gold to physically purchase them. You get 2 Command Points for 1 Gold in Avengers Alliance. You also receive 5 Command Points when achieving a 5 Star rating on any level.
Sometimes during a fight the enemy will drop a Command Point. Upon levelling up, you also have a chance of getting 1, 3 or 5 Command Points.
Note that it seems as though the more Command Points you spend recruiting Heroes, the higher the cost of them gets in terms of spending Gold on them. This means that if there are Heroes that you particularly want to get, try to get them early so as to avoid the price hike later on.