Grab both sides of the back of the snake's head and you should notice that the mouth opens up a little bit. After that, get a Q-tip and put the middle of the Q-tip at the front of the mouth and go in. After you are in the snake's mouth with the Q-tip, pull the Q-tip downwards to open up the mouth. Make sure the cotton on both ends of the Q-tip does not touch the snakes mouth or teeth because the cotton can get caught in the mouth and the teeth.
Here's a YouTube video that shows how to do it: YouTube - How to check for a Resperetory Infection.