USB Audio Interface
To connect your computer to your Hi-Fi you will need a good quality USB Audio Interface (digital-to-analogue converter). There are many available with prices ranging from $10 upwards.
If you want to sample your LPs, you'll need one with an Analogue-To-Digital converter and a phono input with equalizer for your turntable, and an AUX input if you want to connect a cassette or reel-to-reel tape deck. If you amplifier has a digital input, you may want to use your amp's DAC, so you need a USB-to-optical converter, which is cheaper.
Here are three examples which will do the job...
Low-cost USB/Audio Interface with built-in Phono preamp/equalizer for digitizing your vinyls, switchable to Line Input for everything else. A good choice if you don't have a "real" Hi-Fi ;-)
Phono inputs allow the DS-DAC-10R to connect directly to your turntable, and there's a ground terminal for it too. The input jacks also support LINE level signals, such as from a cassette deck or reel-to-reel. The sound quality is pretty good. I use this one, with the MuMan Suite replacing the rather limited Korg AudioGate software.
If you also play guitar or sing, then a USB audio interface with guitar and microphone inputs could be a good solution. The line outputs are via two 1/4" jacks, so you might need an adapter cable to connect to the phono inputs of your Hi-Fi amplifier. Or you can connect directly to your guitar amp, or two amps for stereo, if you really want to annoy the neighbours. To sample vinyl, you can connect you Hi-Fi amplifier's Line outputs to the Line inputs of the UR22 using the jack-to-phono cables.
If you are insanely rich, then you could afford the insanely expensive Chord Hugo 2. With this you can play back at up to 768kHz (!?), but you can't record anything. Even so, your neighbours will still be really jealous. This one wins all the awards, and it's made in Britain.