I have made several characters, but they've all been in the sitting position. What do you think is the best method for making them stand?

I would stick a skewer through the figure and let it dry like that. Also make sure it's long enough to reach the bottom of the tier.

I use wooden skewers as suggested above, but also make them in stages allowing drying time in between (ie, make the legs, let dry, add the torso & neck, let dry, then add the head)