Alivio 8 speed rear derailleur cable broken

There is a bolt on the derailleur that holds the cable. Loosen it to remove the cable from the derailleur. If there is a cap on the cable, remove it by pinching it with a pair of pliers. i do not know what style shifters you are using, but most all of them have a plastic cap covering the port for the cable. Shift to the highest gear to prep the cable for release and remove the cap. Push the cable back into the shifter and it should come out of the port you opened. Now, slide the new cable in and pull it through its circuit of cable housings and back to the derailleur. Attach the cable and shift to the lowest gear (big cog). Now, pull the cable along the top tube or seat stay (wherever it is more accessible) away from that tube very firmly several times. This pre-stretches the cable and minimizes necessity of adjustment later. The harder you pull, the better. Once you have stretched the cable, shift to the highest gear again and release the cable on the derailleur. Adjust the barrel on the derailleur to the center of its range and turn in the one on your shifter all the way. Now, pull the cable taut and reatach it on the derailleur. You can cut the cable to length and put a cap on it now. Shift into your middle gear and use the barrel adjuster on the derailleur to align the jockey wheel over the appropriate cog of the cassette. You should be all set!

i hope this helps
-matthew