The simplest way may not be the hi fi way but,
1. turntable connects to phono preamp input--The phono preamp contains RIAA equalization curve, without the curve of equalization your records will sound wrong.
2. phono preamp output connects to adapter. The adapter changes RCA plugs to 3.5 MM stereo plug male.
3. Adapter connects to the line input of the computer. Make sure the input you are using (on the computer) is set to LINE level, not microphone level. You can change this level with the audio software, provided by your motherboard or computer manufacturer. (your audio software is probably already loaded)
4. Install an analog to digital converter application, such as Sony Sound Forge.
Once you have played and recorded the LP, you can save the file as: MP3, or other format.
The recording may be played with windows media player, etc...or you can save it as a regular CD audio format and play it on a conventional CD player.
TIPS: Your phono stylus and tracking weight are important factors in sound quality. A crappy stylus = a crappy sound transfer. Incorrect tracking weight = no bass frequencies.
Of course you can use another method besides RCA to stereo plug, you could buy more expensive equipment such as USB turntable, etc...
you could use free software, but I don't recommend it. Sound Forge is a studio quality, bit rate converter, etc...which will come in handy for many audio projects.
DANGER: do not attempt to connect the speaker outputs of an amplifier to the computer line audio input. It melts too easy.