After every game, you'll see a list of the challenges that are available, under 4 different categories. You can have a total of 4 challenges active during any game, assuming that you are in a squad, only 3 active challenges if playing alone. They seem to be a little random, and range from extremely easy to extremely difficult. Just make sure to pick the easiest challenges for you, and try your best to complete them in every game. You'll have a few chances before the challenge expires and changes to something different.
Also, it should be noted that you can do the same challenge multiple times, and it will count, as long as it comes up again on its own.
Note: Sometimes you'll be given challenges that are impossible considering your level. They may ask you to use a certain weapon that unlocks at level 30, when you are only level 15. Don't worry about these, just ignore them for as long as you can, and they'll be replaced by something else.