You basically need the part of your fingers from the knuckle (closest to the tip) down to be straight up and down. You accomplish this by arching your fingers and really bending that first knuckle towards the tip. Work on consciously bending that first knuckle on each finger. The more you play guitar, the more you'll have control over this.

When I first tried to do this I thought, "Man, is it always going to feel this weird and uncomfortable?"

And I can tell you after a year late, the answer is "no!" It's very natural for me now. So just keep at it.

What I would do first is: forget about your first finger and just put your middle and ring fingers where they need to be without accidentally touching the g string. Once you have that down, then add your first finger. Then lift up all your fingers at once and put them right back down in the C chord position (very slowly at first) so you can work on going right back to the right spots.

Good luck!