Your turntable must have a "Line Out" or "output" jack that can be run to your computer. You will also need cables to connect the "Line Out" on your turntable to the "Line In" on your computer. Most computers use a simple 1/8 inch stereo jack for input; if your turntable uses a different type of output than your computer's input, make sure your connection converts one to the other, either with a special cable with one connection on one end and one on the other, or adapters.
You will also need a computer with a sound card and audio editing software. Audacity is a free software that can be downloaded from the Internet, but there are others that give you more control. You also need working speakers or headphones.
Open the editing software and do a test play of the LP; you should be able to hear the music on your computer speakers and/or see the signal displayed on a graph in your software. Refer to your help and user manuals if necessary. Set input levels so they are strong but not peaking.
When you're ready, click the "Record" button on your software display, then start the first track on the turntable. When the track ends, click "Stop." Repeat this process for each track you wish to record. Always start and stop recording on your computer before starting/stopping the turntable to avoid missing any of the track. When each track concludes, save the file on your computer as a "wav" file; you can convert it to mp3 when you are done editing.
The software should have a function to "clean up" the hisses and pops from the recording. Open each track on your computer and use this function if available. You can also remove the silences at the beginning and end of each track. Don't save your changes until you're happy with them--or you can save a copy of the file you're playing with, in case you want to work more on it later.
When you're happy with the track, use your software to save it as an mp3 file. Follow the instructions provided by your software provider, and refer to the "help" section if necessary.
You assume all risks and liabilities of violating copyright laws, even when copying LPs that you own. According to the United States Copyright Code, there are "fair use" exceptions for making copies, but this term is not well-defined. There are also certain permissions given for libraries and "archives" to make copies for non-commercial purposes. As a rule of thumb (not legal advice), only make mp3 copies of LPs you own, and never sell, give away or lend copied material to others.