The quick and easy way is to connect the analog audio outputs from your DIRECTV box to the line-in socket(s) on your PC, record with
Audacity (which is free but every bit as good as any other consumer level recording software) and save the recording as MP3/OGG then copy it on to your Axim for later playback.
If you can be more specific about your current setup and requirements then better advice can be offered.
What kind of DIRECTV box do you have?
Does it have digital audio out?
Does your PC have a sound card with digital audio in?
Is this a maximum quality for minimum cash deal or is it OK to spend a few bucks?