I have many years of building and remodeling behind me but I have a strong inclination to do plumbing work. So, the reciprocating saw is a favorite tool for many applications.
After you cut the rotten part out, you can replace it with a new piece, fill it with wood putty, sand it smooth and paint.
However, you may run into more decay than you can easily repair and want to replace the whole door frame. It is a little more of a challenge but it gives a lot of satisfaction. Here again, you can take the casing and brick mould off and cut the fasteners with the ol' recipro.