I've never done this before with any handgun because if i wanted the tactical model i've just bought that model...
But i acquired a new HK .45C for a really great price & would like to buy a threaded barrel for it.

What i don't understand is, how it is legal to purchase an HK threaded barrel & use it on my weapon when the serial number is on the original barrel that came with the gun?

Thanks!

It is completely legal. While the frame, slide, and barrel on your 45c share the same serial number the "firearm" as far as the ATF or any other .gov is concerned is the frame only. Changing barrels is not an issue. I have a 45ct and did the reverse and bought a regular barrel for carry duty. You are good to go.

ETA: I live in Alabama. This might not be the case if you live in a state that requires those stupid fired shell casings.

The serial number on the barrel is an HK thing, and is not a legal requirement.
Many people do exactly what you plan on doing with no problems.

Glock does this also. I'm sure other manufacturers do as well. I don't know why but as JBnTX said, it is not a legal requirement.