The computer control port on the TS-2000 is the RS-232 port on the back of the radio. In spite of what you hear about RS-232 to USB converters, there are many available that work just fine. You need to find one that uses the FTDI chip set and it should work OK for you. (There was a problem with an updated driver that Windows Update was pushing out to Windows 7 computers a few months ago, but Microsoft and FTDI have addressed that.)
More important than the model of your computer is the version of the operating system that computer runs. If it's running Windows 7 and you are satisfied with it, then there's no compelling reason to change.
The real question is what do you want to do with this computer-to-radio connection as that drives the software you choose. Many people will suggest Ham Radio Deluxe, but then they tell you to get the "free" version from some unsupported web site. The free HRD is an old version that hasn't been updated in a couple of years though many people are still using it. The paid version of HRD is a very good radio control, contact logging, and DX spotting program. Your payment gets you a year of very good tech support from the HRD folks. After a year, the program keeps working, but you don't get free tech support unless you renew. If you pay attention, you can get a discount on HRD usually during major hamfests. I have used the paid version of HRD and found it very capable. DM780 is the digital modes module of HRD and I ran into a limitation with the way DM780 worked, so I have since abandoned HRD.
Another very good program that does radio control, contact logging, DX spotting, and other features is DX Lab Suite. This program is free and the author is very engaged in supporting the program. The DX Lab Suite modules aren't as pretty to look at as HRD, but they work very well.
Both of these programs integrate pretty well with ARRL's LotW.
Those are the two radio control and logging programs I'm familiar with. There are others out there that I've looked at, but those are the two I've used extensively.