You will need a cable to connect the hifi headphone socket to the pc's sound card. You will need to check which type your systems have but most use 3.5mm jacks ( older hifis had 1/4" headphone sockets). See here for typical cables you can get them from Maplins or Currys / Dixons / Comet / local Hifi shop. 3.5mm jack cable
http://www.maplin.co.uk/Module.aspx?ModuleNo=1 1422
Then you will need audio rerording software Audacity is very good and free to download and use. It has an extensive help facility as well as a useful faq
http://audacity.sourceforge.net/
Once installed and you are ready to go.
You can save in either wav (uncompressed) or with an extra codec (Lame) mp3 (compressed) mp3s will take up much less space typically 1/6th depending on your compression settings. Most classical music is good at 192kbps, some rock music is ok at 125 kbps. Below this level you will start to hear all sorts of artefacts so 125kbps is the minimum.
It helps if you record each track one at a time rather than a side at once. Once saved on your hdd you can either burn as mp3s (playable only on computers and some modern hifis) or as an audio cd using Nero or similar.