Yes, except for the fact that all the drivers on your old computer are not likely same as new one.
1. Disconnect HD cable in new PC
2. Yank HD outta old PC, install in new PC.
3. Format and do fresh install of OS on old HD in new PC
4. Disconnect old HD, connect new HD and boot.
5. Connect old HD, go into BIOS and pick what HD you wanna boot from by changing boot order priority