Hello Matt,
Open up your computer and usually down the bottom you should see your hard disk bays. Remove that hard disk (sometimes clips, screws, each computer is different) and the hard disk should pull out. Note, you may have to pull off the two cables first. Turn your hard disk over and look for a big label (usually white and on the bottom for Seagate) and there should be your serial number written on it. Note, if the label is not visible, then you may have to remove the plastic/metal casing around the hard disk. This IS NOT opening the hard disk, just removing the plastic that secures it to the hard disk bay. If you have trouble, just ask and to see what your label should look like, see here:
SerialNumberLocator
Richard