Sadly I don't think you can upgrade in the demo,but I've gotten just over 2,000 exp. Here's what I did, aside from winning fights with flawless combos:
1. Take out the thugs Joker first sets on you, pull off the grate, get inside, and get the Riddler Trophy. 2. Go to the room Joker used to set the thugs & hit three sets of teeth with batarangs. A fourth set is just down the hall. 3. Take out the two thugs who attack you, go left to take out another set of teeth. 4. Take out Zsasz, explore the room and rip off a grate in the front end of the room on the opposite wall for a Trophy. 5. In the final room, take out all but one thug. Go to the exit for that room, the door with graffiti.Only do this if that last thug is on the opposite end, so to as avoid a fight. There's three more teeth to thrash. Now you can go back and take out the last enemy.
"Nothing is real if you don't believe in who you are!"- Rocky Balboa http://mmtrigger.deviantart.com/
Experience isn't set for combat either. I've beaten the first battle and gotten around 75 experience. Later, I fought those guys and got a really long combo string and used a variety of moves and some multipliers kicked in. I ended up with 365 points. So given the above checklist of things to do plus all the things that might multiply your score, you might be able to end up with more experience than you think at first. No idea if it's enough to level up though.
If you're gonna scream, scream with me. Moments like this never last!
Yea I knew about the riddler trophies, did you know that there is a question mark symbol without the dot on the window inside the building in the silent predator bit? You gotta have detective mode on to see it, the dot of the question mark is on the opposite side of the room, you got to piece together the 2 to make it look like a question mark, but nothing happens in the demo though, but I bet you gotta scan it or something in the real game.