You cannot put the old drive in a new system and boot directly from it using the old OS - not only do you have an activation issue (and you can't use the Dell in a non-Dell system) but drivers most likely will be wrong preventing the OS from working.
The easiest way to access the old files is to put the old drive into a different system that already has a HD with the operating system, using the old drive as a secondary drive, and then you should be able to read the files and transfer them if you want. The old drive will get a new drive letter in the new system - something like E of F - depending on how many other drives you have - and you access files, store, and copy files the same as you did on the C drive.
If you don't plan to use the drive back in the old system, you could erase all the old files you don't want and use it in the new system permanently. One good way to clean off the old files would be to first transfer all files you want to save to the other drive then reformat the old drive. You could then copy back any files you wanted to keep on that drive.