If you have a light deodorant stain that's pretty minimal, or you're in a rush and trying to head out of the door as soon as possible, I have a quick and easy fix. All you need is some tights or knee-high socks.
You'll want to take the piece of clothing and lay it on a flat surface. Pull the tights tight in your hands, and rub it briskly against the stain. It doesn't have to be a heavy rub, just make sure that it's a brisk motion, with a flick of your wrist. Continue to do this until the stain is gone. This should be able to get out the lighter stains and is extra helpful when you're on the go.
In order to get rid of larger, tougher deodorant stains, you'll basically want to "flush" the stain out of the clothing. This looks like flipping your shirt inside out and running cold water right against the mark. You can use your fingers or grab extra pieces of fabric and rub it against each other in order to really get rid of all of the deodorant residue. Then, mix bicarbonate of soda with room-temperature water until it forms a thick paste-like consistency. You'll want to use a spoon to apply the paste directly on the stain and allow it to soak for about 20 minutes before rinsing it off with hot water.