How deep should my grout removal go?
To get the best results you should remove down to a depth halfway so that the new grout can bind to the original grout and to the edges of the tile.  Removing too little will not last long since the new grout will separate.  I find the best way to remove the grout is to use a dremel with an attachment that will remove the grout.  see link to video.

Dremel Grout Removal