I had the exact same scenario and did it myself.
If you aren't handy, do what bunny22 says and hire a good sheetrocker (really any good contractor).
These are the steps that I took:
1) Keep the initial sheetrocked wall up. If it's brick, you'll need to add a first layer of sheetrock.
2) Get Green Glue Noiseproofing Compound
3) Get the thickest sheetrock sold at your local hardware store (typically 5/8"). I got the type X which is firecoded. You could do what bunny22 mentioned and get quietrock which will help, but most contractors don't want to deal with it because of how heavy it is.
4) Apply the Green Glue to the back of the sheetrock (go with the highest amount that they recommend).
5) Hang the sheetrock in the opposite direction that the first layer is hung. Typically the first layer is hung horizontally, so you would hang the second layer vertically. This is to avoid seams lining up and allowing some sound leakage.
6) Get Green Glue Noiseproofing Sealant
7) Seal seams between sheetrock and the top and bottom of the sheetrock.
8) Mud and sand as usual.
Give the Green Glue 30 days to set completely. At this point, you shouldn't hear your neighbor anymore (unless he just cranks his bass).
If you do hear some noise, make sure you didn't forget to use the sealant to fill around electric boxes and any other possible conduit for noise not covered by the Green Glue.
So basically the only part normal contractors aren't doing day to day is the Green Glue portion, which really is dead simple.