Did you take a small screwdriver to the two small screws on your rear derailuer? One problem could be that your stops are over adjusted and preventing your deraileur from moving the entire distance.
Another problem could be that your new cable is too short preventing your derailuer from shifting into 5th and 6th.
Check out this site they have some basic tutorial videos that should help you out fixing your bike.
Good Luck http://bicycletutor.com/