There are many cheap ways to get this done, like the afore mentioned big feet technique, but there are other, more time-consuming ways too.
If you are proficient at photoshop/ photo editing (any photo retouching program will do), you can build a rig with a very heavy base (like a block of lead), that has a rod coming out like an arm - you can use this to hold the character without taking up too much screen space - just editing out the dowel that is holding up the character. For a similar example, see this: http://www.stopmotionanimation.com/handbook/12.htm

The other way, which I used to use, was to go berserk with the character until it would finally not move. . . or hold it, and right when you take the photo yank your hand out of the screen - works really well, with the motion blur of the falling character.

In all seriousness, though, I would recommend building a simple wire armature, and then placing clay around it - easier to control and stand (by 'stand' I am welcoming you to the world of tie-downs - screwing into the sets).