Draw a path from Illustrator in After Effects

Second time this morning. You convert the Vector Layer to a path using the Layer/Create menu. Then you clean up the converted shape layer and add a stroke and trim paths. It is important to have each set of paths you want to 'draw on' on a separate layer and you must have only a simple stroke applied to the path in Illustrator. Brushes, gradients, raster effects, and just about anything else you add to an object in Illustrator will foul up the conversion. If your illustration has brush strokes you can duplicate the paths on a separate layer, remove extras, name layer something like track matte, and then use that layer as a track matte for the drawing in AE to reveal the paths. 

Here's the tutorial I created this morning:

You can also group your paths, add a fill and trim paths to a group,  and re-arrange the path order to get what you want.