If all the data is on your C drive currently, then there may be a conflict if you hook it up into the new machine as both drives will have a version of Windows on it.
If it is on a separate drive to the original C drive then you should be ok just hooking it up.
If the new PC is running SATA Drives then bear in mind you will probably only have one IDE port rather than two so if you attach it to the same cable as your DVD/CD drive then it will run pretty slowly. There shouldn't be any compatability with runing both types of hard drive.
The other alternative as you said is to get an external hard drive case and just plug it in. This usually works pretty easily. I think they're around 40 quid. Useful for backups, but don't really run fast enough to use for anything else.