There should be 2 challenges available at the start. The humans one and a demolition mode one towards the bottom. Do you have the Heroes DLC? If not, maybe that's why you only see one?
Housemarque is aware of the "can't connect to game server" issue that's affecting Challenges. You should be able to do challenges just fine and hopefully our scores are saved so as soon as the rankings are fixed, they'll pop up. Don't quote me on that, though.
So far, the quickest way to rank up (at least in the first 10 levels) is to play a challenge online. Try the default one. If you win AND get a gold medal (or maybe just win?), you'll get 135 CP. Since you only need 100 CP to reach the next level, you can go up a rank every time you win. That's how I'm at Rank 10, atm.