I've imported puppet parts into After Effects CC 2015.
I rasterized the layers, and added the puppet pin tools.
Two things have happened:
Lines in a silhouette shape of my character go through it when I move it around. But when I untick the sunhine symbol, they disappear.
Secondly, using the puppet pin tool, then moving the character around with a null object, messes up the pins, is this because the images are rasterized?
It seems you do not understand how it works. The Puppet Tool is a raster effect, so you can forget about using continuous rasterization. It's just another distortion effect and the same limitations apply as if you were to use any other effect. Since it's an effect, all coordinates are relative to the layer, so conversely, moving stuff around will require a consistent use of coordinates, meaning you will have to apply expressions to convert coordinates and/ or avoid mixed setups and instead focus on doing everything inside the tool. you simply have things backwards. If you need specific help, provide more info and screenshots.