There are two parts to this issue:

Part 1: connecting something that can read records with your computer

For this, you can either play the record on your normal turn table, and take the audio output to your line-in via cable. You'll need a cable with 1/8 inch male audio on one end, and whatever output your turntable has easily available. EG a headphone jack on your stereo. Cables like this are available from audio stores like radio shack or best buy or whereever. If you have more speicific questions about this aspect, I'd be happy to provide more details - comment or edit with more details.

Alternatively, you could use something like @Jim mentions in his comment - a USB turntable. I have 0 experience with these or if they are Ubuntu compatable.

Part 2: record the audio that's coming in. For this part, @fossfreedom's answer here covers this in good detail.