To install an additional HDD in your machine you usually need THREE things: a vacant drive bay, an unused HDD port on the mobo, and an unused power supply connector from the PSU. If any of these appears not available, consider these alternatives:
1. No vacant bay (OP's stated problem). Most HDD's now are 3½" size, the same as a floppy drive. Look for an empty floppy drive bay and mount your HDD there. You don't need to remove the front cover since you will not be slipping in a diskette. If that's not available, what about an empty 5¼" bay - the ones used for optical drives now. You can get adapter rails that mount on the side of a 3½" device and make it fit the 5¼" opening.
2. No vacant HDD port on mobo. IF your device is an older IDE unit, are you sure you have TWO IDE devices on EACH IDE port / cable? If not - if you have only one device on a port, but its cable has no extra connector in the middle - just get a new ribbon cable with three connectors in total. ALL IDE ports can support TWO devices on one cable. Just remember to set up the jumpers correctly to establish ONE Master and one Slave device on EACH IDE port. However, if you are hooking up a SATA device, and you already have all your SATA ports on the mobo in use, consider buying / installing a SATA HDD controller card for the PCI slots. Typically you can add 2 or 4 SATA ports to your machine.
3. No power supply connector? Get an adapter that converts one PSU power output for a HDD into two output connectors, and go from there.
If you really cannot find a bay to install into and want to upgrade your existing HDD to a larger capacity unit. your main concern is cloning ALL of the data from the old HDD unit to the new one. As long as you can connect data and power cables to the new HDD (even if only temporarily by disconnecting some other device during the cloning work), you do NOT have to physically mount the new HDD inside your machine to make the clone. Connect it up, then set it on some simple insulating surface - even a sheet of dry paper is enough - that is secure enough it won't fall down. Leave your case open for the cables. Use cloning software to make the clone to the new HDD. Then shut everything down, remove the old HDD, mount the new one in its place and connect it up.
Free is good! IF you buy your new HDD from Seagate, go to their website and download a free utility from them, Disk Wizard. It is actually a customized version of a very good package, Acronis True Image, which does a LOT more than just cloning. So make sure you get the manual that should comes with it and read it. Your best bet is to install it on your old HDD, then run it from there to make your clone to the new HDD. Note that it does not care what old HDD you have, but it will only make the clone copy TO a HDD from Seagate.
A clone made with this tool will contains a complete copy of EVERYTHING from your old HDD, with your OS located in the right place so you can boot from it. The whole idea is that the clone on the new HDD will completely replace your old one so that your machine behaves exactly as it did before, except that your new HDD has much more Free Space on the C: drive.
Now, IF you buy your new unit from WD instead, go to their website and download their customization of the same stuff, called Acronis True Image WD Edition. Again, it only makes clone copies TO a drive from WD.
Other HDD makers may have similar free cloning utilities. check their websites before buying.
A couple hints before you make the clone. The first steps the utility does is to create a Partition on the empty new drive and Format it. These are steps that MUST be done with any new HDD, and the software makes it easy. It has default settings it wants to use and goes through screens to let you confirm them. BUT some I've seen make the new Partition the same size as the old drive, which is usually NOT what you want. After all, you're moving to a bigger space! So, check through the menu system (that is why reading the manual ahead is helpful) and find where you can over-ride the defaults and have it make the new Primary Partition the full size of your new HDD - I presume that is what you want. By default it should plan to make this Primary Partition a bootable one which you want, but just check that. You will want to Format with the NTFS File System, the default setting. You can choose between Quick Format and Full Format. A Quick Format does all the necessaries in about 15 minutes. A Full format does that, then exhaustively checks ALL of the drive's sectors for errors and notes any it finds as Bad Sectors that Windows will never try to use. That process may not be necessary with a good new HDD, but it is a good precaution. However, it will take MANY HOURS to do that job, so plan to let it run and do something else in patience.
Once you have the parameters you choose customized you tell the software to run the job, then wait. It will Create the bootable Primary Partition, Format it, then copy absolutely everything over. When it is done I suggest you exit out and shut down completely, then disconnect power. Remove the old HDD and connect the new one to the cables that were on the old one. If you had to temporarily disconnect something else for this cloning job, reconnect. Close up, reconnect power, and turn it on. Your machine should boot completely normally. As I said, the only difference should be that your C: drive will have a bunch of new Free Space.