Avaunt means "hence", or "away (from here)". It is derived from the same roots as the French avant, meaning before, or on ahead of.
It is is used as a command in several Shakespearean scenes, similar in sentiment to "Get thee hence!" , often said to witches. In spirit it is not unlike the expression "Get thee behind me, Satan!", except in the other direction.
Avaunt is not related to avail, which implies function rather than distance.
A bodkin is a type of pointed arrowhead, an advance on the broad arrow, and is also a tool for punching holes in leather, a Renaissance hair-pin, and a slender-bladed dagger. To bare one's bodkin is, presumably, to unsheathe a dangerous blade.