As far as the connection to the computer, if it is a desktop computer and has a sound card, then there are headphone, mic and line jacks. Depending on the age of the system and the type of desktop unit, these may be on a card which has connectors on the back of the computer. Some models had the sound integrated on the mother board and in the case also, so one has to look for the green, orange/red and blue/purple 1/8" phone plugs, either on the case or on the various connector panels on the back of the system. Some sound cards may not have a line jack, which is usually the blue/purple one. There will be sikscreened or engraved icons next to each jack.
On a laptop, there will be a mic and headphone jack, should be color coded 1/8" jacks somewhere on the side of the computer.
As far as the program, I know of Audacity which is a GNU license freeware program. I use Goldwave, which is also good but is shareware for $41. It would do fine, but I use it to dub from a DAT recording deck. Audacity has similar features to the software that strumminsix mentioned.
Main thing is you need a decent mic to get any sort of good sound. A mic w/ enough dynamic range is required to get the bass you are missing. Put your money towards a mic, provided your computer has a sound input jack. If you already have an acceptable mic and can't do the computer, recording machines are the most common things on the market and they are in the low hundreds. Also a cassette deck records the tape too slowly to get really good sound from a mic. Hard to explain but true.
Hope this helps. Good luck with it.
Dave
The river keeps a talkin'