Hey, can I suggest something? If I'm reading correctly, your main goal is to be able to play records, not necessarily construct an audiophile's dream. The main point here is that you need three separate components, a turntable, an amp, and speakers. It doesn't need to get complicated.
Since you seem very focused on scrimping cash, I'd focus on getting a good turntable. You mentioned having computer speakers -- well, they suck, but they'll play music, and they have a built in amp. You can connect the turntable to the computer speakers (there's no reason you need an actual computer involved in this) and play your music.
Later on when you want to spend more on your system, you can upgrade to an amp and better speakers, at which point you'll be glad you spent your initial money on a good turntable.