So I've made a new char thats level 10 now. I wanted to make a good enchanter, but I'm having trouble finding gems.

I did manage to get my self a Soul trap enchantment on my weapon, but due to the enemy not dying in time the weapon got empty. Does the soul trap spend a charge on every hit, or just every 3-4 seconds?

Anyway the weapon got depleted and thus I couldn't fill my gems. I have a lot of petty and common souls gems that are waiting to be filled, but since they all are empty I can't recharge the weapon with soul trap.

Of course I could buy a charged gem from Farengar but I don't wanna buy it since they are overpriced. I don't do pwerlevel on this char so it's not a very rich one either.

Where should I go hunting for charged gems?

Firstly, yes it spends the charge each time you hit, secondly instead of using your soul trap weapon as your main weapon only use it to last hit the creatures so you only spend one or two charges finishing them off. As for charged gems you can find them mainly on wizards, friends and foes. If you don't mind a little criminal activity you can go to the College and steal a few. Otherwise you'll have to wander through dungeons with wizards in them until you get some. It shouldn't take too long if you can find wizardly dungeons, the only thing is, depending on your difficulty setting, you might not survive against a wizard. A couple dungeons with a good number of wizardly foes in them are if you want to find them, and don't mind more spoilers, you can go on the Skyrim Wiki and find where they are.

Dwemer automatons usually drop soul gems. They're not always filled, though.

Definitely be careful, though, since Dwemer Ruins are pretty tough at that level.