I'm not sure what you mean by "converting", but basically it would be like this:
Wall socket/Antenna >> Coaxial cable >> Freeview STB >> HDMI cable >> Monitor
If I understood your question correctly, then the Freeview set-top-box is indeed the one doing the conversion process. And even if you attached the STB to a TV, then it would still be the STB doing the conversion, (unless you plugged the coaxial cable straight to the TV, of course).
If you went with a USB TV tuner, then it would be like this:
Wall socket/Antenna >> Coaxial cable >> USB TV tuner >> Computer >> HDMI cable >> Monitor
But in any case, if you haven't yet bought anything, but intend to purchase a new display device for the computer, and don't mind going for a bigger screen, then you should definitely give a thorough consideration for an HDTV.
There is no hassle with those, and you won't need a separate tuner/box (or any other device, for that matter) for it to work, as the tuner is already built-in inside the TV. In such case, your setup would be like this:
Wall socket/Antenna >> Coaxial cable >> TV
And you can additionally use the HDTV as if it were a regular big PC monitor, as well.
BUT:
If you have no immediate need for a new display device, then the USB TV tuner is certainly the cheapest way out.