While I don't really know much about Illustrator, this image doesn't seem to be an ideal choice for converting to vectors. I'd definitely have to agree with Jolt: if you just need to sharpen it up for printing, there are lots of quick and relatively easy ways to do that in Photoshop using either the built-in tools or third-party plugins like
FocalBlade.
However, if you really want to go down the vector route, I suppose you could try this Rotoscoping technique:
Imitating A Scanner Darkly | Illustrator Techniques