If you're getting a new PC immediately with its own new HDD and OS installed, there's a real easy way. Just remove the HDD from old machine and install in the new machine as a second drive. Good idea to check that its BIOS is set to boot from its own new drive and NOT from the second unit so it will ignore the old OS. Then you can just leave the data on the old drive and use it. Or if you want, you could copy all the old data to the new drive, then re-partition and format the old one as a plain data disk (not bootable).
But your post suggests you are not getting a new one right away, and need immediate and ongoing access to the data on it. You need to convert it to an external drive you can carry around and hook up to any other computer. That's easy. You buy a new external case, move the HDD from old computer to case, and voila! I will assume your old HDD is a 3/5" either IDE (PATA) or SATA or SATAII. Here are tips for this:
1. There are two interfaces you need to specify. One is how the HDD plugs into the case, and that is determined entirely by your old HDD. If it is IDE (PATA), get a case that accepts this type. Make sure the old drive's jumper is set to Master and install in the case. If old HDD is SATA or SATAII, get a case for this and move the drive. Ther are no Master or Slave Jumpers to set on SATA drives.
2. The other interface is between case and computer. One of the best (fastest) is eSATA, but it requires that the "other computer" have one of these ports built in, and many don't yet. But almost all have USB2 (or at least USB on older comps), so be sure you get a case that has this. Most now come with at least two possible connections - USB2 plus eSATA or Firewire400 (aka IEEE1394a).
3. To use the extrenal drive with another computer, you may have to install on the comp a driver so it can use this new device. The case should come with a CD of softare and drivers, so keep this with your unit as you travel, in case you need to do the driver install on a new machine you hook up to. Almost guaranteed the "other computer" would never try to boot from the external drive, so the presence on it of an old OS is no problem.
4. Get a case with its own power supply. It may be in the case, with a normal power cord. Or it may be a "wall wart" type. Pulling enough power for a 3.5" drive from a USB connection can be tough. Most external cases now come with their power supplies, anyway. Keep this with you as you travel with the external drive.
5. Some people prefer to get a case with a built-in cooling fan to keep HDD temps down. Some argue it's not necessary and will eventually wear out, anyway. Depends a lot on how hot your HDD unit gets in normal use. Your choice whether to get one with or without fan. With fan costs more.
6. When you use the external drive, ALWAYS check for and use the "Safely Remove Hardware" icon at the bottom right. Never unplug the external HDD without using this to "disconnect" the HDD first. Doing it properly makes sure that any pending writes to disk are completed before unhooking. Failing to follow this rule could leave you with corrupted files and a lot of grief.