After you have completed your placement matches you simply have to continue to play games. The result of your match, a win or loss, is what determines your rank. You will not rank up after a set number of wins however you will rank up once you reached the required Matchmaking Rating for the specific Rank.
Players receive Matchmaking Rating points for winning PUGs and lose Matchmaking Rating points when they lose PUGs. The amount of points you win and lose per match depends greatly on a variety of factors such as the size of your premade group, the balance of the match, and more.
For example, if you are Rank B you will see a numeric value between 3,500-3,999 on your profile. This value represents your Matchmaking Rating and shows how close you are to ranking up or down. All players will stand to gain more by playing similarly skilled opponents as competitors will be rewarded MMR proportionately based on a number of factors such as the MMR difference between you and your opponents, your group size, your opponent's group size, etc.
Your Rank will also decay if you are inactive as seen here.
Below is the current Matchmaking Rating Chart
