You can play normally and you'll eventually get to level 55. However, to speed things up you'll want to take advantage of Double XP weekends when they occur, and more importantly the available challenges.
When you're in a multiplayer menu, press the
button on your controller and press RB to go to challenges. Navigate to the "Multiplayer" section, and you should see a screen showing those challenges which you are nearest to completing and overall progress. Each challenge you complete rewards you with a nice bunch of XP - have a look around this section of the menu and look for challenges that you think would be easiest for you and go for them in game. This might mean you need to frequently switch game modes, specialists and weapons to earn as many as possible. In most cases, you'll earn challenges through normal play, but they'll become less frequent if you don't switch things up.
To keep it simple, just take notice of the challenge which is listed as "Nearest to completing" and aim for that. Once it's completed, you'll see the next one to go for.
As KILLER1NST1NCTS suggested, consider jumping into Domination mode to start things off, since it is one of the quickest way of getting the objective type challenges.
Have fun!