G'day,
I've been doing this for about 3 years now.

Here are the main steps:

1. Record on your computer while playing via your amplifier (Phono in, Tape out, thanks Alex, see below) into the LINE input of your sound card, not the microphone input.

2. Use your sound editing software to normalise (amplify to max level without distortion), BUT listen to the results, some songs may end up too loud.

3. Use whatever software you have to remove clicks & pops. Again listen to results, none of these do a perfect job. If you don't set them up right you may introduce unacceptable distortion.

4. Optional: do a noise reduction; reduces surface noise and noise from the original master tapes.

5. IMPORTANT Save each song as a separate file. Make sure there is some silence at the beginning and end of each song. Commercial CDs often have about 1/2 sec at the beginning & 2 sec or more at the end. Adjust to your taste.

6. VERY IMPORTANT Burn the CD in Disc At Once mode, NOT Track At Once.

Notes: Typically, an amplifier has a pre-amp section designed to accept a phono input from a magnetic cartridge. This provides both gain and frequency compensation according to the RIAA standard (briefly, bass boost & treble cut). Without it the signal will be much too low level, it will also severely lack bass and greatly accentuate the treble. You can normally utilise this by taking the tape out from the amp to the line in on the sound card.

Sound editing software normally offers several choices of file formats, e.g. mono/stereo, 8 bit/16 bit etc. but note that there is only one choice suitable for CDs: stereo, 16 bit, 44100 samples/sec. Use no other.

The software I use for all my audio recording/editing/processing is Cool Edit Pro. It has all the functions you need including click reduction, noise reduction etc. It may be overkill for this task though and a bit expensive. It's also a multitrack recorder if you want to set up a recording studio.

For the really technically minded, another trick of mine is to use a modified RIAA preamp. I've removed the treble cut portion of the circuit & record via this. Of course there is too much treble at this stage, but it also results in clicks, pops & noise being accentuated. Now it is easier for the software to find & reduce the blemishes. I then use Cool Edit to provide the treble cut portion of the RIAA curve which it does very accurately. The result is better noise & click removal.

Cheers,
Alan