To actually answer the main question, yes you can level up in sPvP; however it is slightly different in that you “buy” your level. When you play a match in sPvP you gain glory for winning or losing. This is considered PvP currency as you use it to purchase from Glory Vendors in the Heart of the Mists. At one of the glory vendors you can buy a tome of knowledge for 1,500 glory and 20 silver. A tome of knowledge grants your character 1 level up. For losing a match in sPvP you get ~125 glory average (sometimes less, sometimes more); for winning you get ~200 glory average. Each match takes anywhere from 5-15 minutes. At 1500 Glory a piece it will take roughly 8-12 matches to “gain a level”. So worst case on average would be you lose 12 matches that last 12 minutes each. That is about 2 Hrs 24 minutes of nonstop sPvP matches for 1 level. This may seem slow, but remember that it is A) worst case as you are likely to win as well and
you are gaining PvP rank points that apply to your account. So really in doing this you can kill 2 birds with one stone. So yes, you can level your character strictly though PvP at the rate of ~1-2.5 Hrs/Lvl.
~Cheers