I bought a slingbox and installed the software on my notebook. It works perfectly.

I'm moving to South Africa for 6 months and keeping my SKY subscription together with the slingbox intact in my UK residence.

Is it possible to record a programme from the signal I get from my slingbox to my PC? Is there a piece of hardware or software I can use?

There is probably several differnet ways.

Probably the easiest is a stream capturing software.

http://all-streaming-media.com/record-video-stream/screen-capture-software.htm

I've tried cam studio on my lan and it works really well. It's easy to use too. Pretty large files because I max out all settings but you can play around with it. And the best thing is it's free!!! Of course, always donate a few bucks to the developer if you use and like the program for all their hard work.

You'll have to adjust some setting like
-region (what part of the screen you want it to record)
-video option to max resolution.
-record audio from speaker.
-etc.

I hope your sky connection has a very fast upload speed. Have you tried the remote access with the slingbox? It requires a pretty fast upload speed to send the video to the internet and get a decent feed.