Instead of going through all the trouble of wiring another doorbell I recommend installing an Extend a Chime instead. This is basically a small transmitter that's attached to your existing chime upstairs and a reciever box that plugs into any receptacle on your first floor. Then when someone pushes the doorbell the reciever will ring letting you know someones at the door.
The transmitter can be programmed in various chimes too that will notify you if it's the back or front door. I installed on in my basement cause I couldn't hear the chime from upstairs if I was down there and it works great. They sell them at your local Menards store or order from the link. Just make sure you buy the type that plugs in directly to a receptacle cause they do sell the type that is battery powered and I don't recommend them. The AC powered is much better. Take a look at the link below to see what I'm talkin about.
http://www.smarthome.com/5081.html