IMO if you're going to do it, do it right and install a socket. You'll probably want to upgrade the bios at some point after you install this. The radio shack irons are fine, just don't get the low wattage ones - get a regular iron and a solder sucking one (with the red bulb). They sell sockets there too.
There's not a lot of solder on the pins to make a good seal for sucking the solder, so I would recommend adding solder to each pin one at a time as you go along, that way when you put the solder sucking iron on there (make sure it's very hot) you get a good seal and pull up all the solder. Depress the bulb BEFORE you put the iron on the pin, you don't want to blow solder all over your board. Be patient, make sure you get all the solder from each pin and that you can wiggle the pin freeely. If there is still solder left you can pull up a trace when you remove the old bios. It's repairable, but really bad. Again, patience is key. When you remove the old chip, you want to make sure to pull it evenly from both ends and do it slowly to make sure nothing is being pulled up on the solder side. Reapply heat as needed to make sure it goes smoothly.
You should read some basic soldering tip guides first to be familiar with proper technique as well.