As of this writing the max profile level is 215, but historically that level has increased whenever new Awesomenauts and shop items have been added. Seeing as how this game released so long ago, I don't imagine there will be much continued support at this point, but be aware that it is possible it could increase again.
Profile experience is awarded in four ways...
Game Session Solar: Usually 10,000 - 20,000 XP depending on your performance
Win Bonus: 7,500 XP
Playing Random Character: 1,250 XP
Playing Splitscreen: 2,000 XP
To give you an idea of what is involved in reaching the current max profile level of 215, I had played for about 26 hours across 175 matches. Granted, a few of those hours were spent idling to determine if that was a viable method for accumulating progress toward the Solar collection achievements... it isn't.
Since you'll be nowhere near the 1,000,000 Solar needed for the final achievement -- I was only at 11% at the time I unlocked this one -- I wouldn't bother playing with random characters or in splitscreen for those negligible bonuses. If you're going for the completion, just pick whichever Awesomemauts you like playing with and try to enjoy the long road ahead as best you can. It's a doozy.