The easy way.
If you sound card in your PC supports duplex mode then you can grab any software that records, I use Audacity as it's totally free. Just tell it to record any sound the PC is making, and play the video. You do, of course, need to mute things like email or skype as it records any sound your sound card is making. If you are recording something that is a live broadcast and not on replay you can still use this method, and if you use audacity you can trim the ends or split the tracks up. The only thing that dictates the length of the recording is your hard drive.
Another way, and my prefered method.
There are special programs that can record streams of audio. I use Replay Music 8, it's not free but I'm sure there is a fixed version somewhere.
Replay Music 8 will capture the audio from 100's of sources and you can also set it to capture raw sound from your sound card.
I've never tried it with BBC I player as I don't use it because I have no TV licence.
But I have used it many times with Spotify, Amazon Prime Video, Netflix and Youtube.
It captures the sound from the selected stream digitally and records it, it does not require you to mute anything.
When you use it with Spotify it will also split the songs into tracks as it records and rename the files with the track names, it also ignores the adverts.