How many missions do I need to complete to upgrade the Operation Breakout badge

It's a custom Valve's variation of the Elo Ranking System and no official documentation has been released.

As missions are quite new, I can only tell from my experience:

How missions work in general

Every mission explains itself in its short description. Most of them are "win a complete match on this specific map" or "get a number of kills in this specific game mode". After you completed this mission, you get rewarded with a weapon drop.

A little bit more in detail

You do have 2 missions at start. Every mission is explained quite simple and in order to achieve it / get your reward you need to finish the map you would complete the mission - so don't leave to early.

For example: "Get 6 kills in game mode Destroy." This means you have to look for a game in this game mode. You will join a server and you likely will kill 6 people. After you have killed those guys, be sure you stay until this map is finished.

In the end screen score board, you will receive your mission reward as a weapon drop. (Since now I only got trash drops, I don't know if they cover up uncommon or rare items, too. All items seem to be out of the new added items, no old ones.)

How to get new/more missions

For this you only have to play one supported game in the finding a game screen. This covers up match making, Gungame, Destroy and so on. In the end of the game, where weapons drop, you will now have an additional chance of a mission drop.

So if you need some missions, you may play a few Gungames, as they're finished quite fast and you get the chance for a mission drop after each game.