I've been playing the bass for about 2-3 months now, and just tonight I finally tried recording some basslines that I've learned onto my computer to see what I sound like. There's a problem, though... when I listened to the recording, it wasn't bad, but there was a horrible screechy/echo noise that completely ruined it. I'm pretty sure it's caused by my left hand fingers lifting off of the frets in such a way that they slide a little and cause this noise to be picked up.

Is this caused by some flaw in my technique? Or is it possibly because I just put a microphone in front of my amp and started recording (I know, I know, that's a stupid idea, but I don't have the proper cable to hook my amp up to my computer right now)? I'm pretty sure it's my technique, as I have noticed afterwards that I can hear the noise coming from my amp while playing normally after not really hearing it before. It just doesn't sound as horrible when I'm in the middle of playing as it does when it's recorded.

I'd also like to point out that it's not the noise of the strings hitting the frets, it's different... and the amp is picking it up.

I'm pretty sure it's my technique, as I have noticed afterwards that I can hear the noise coming from my amp while playing normally after not really hearing it before. It just doesn't sound as horrible when I'm in the middle of playing as it does when it's recorded.
String squeak is a fact of life. It's just something you've got to minimize as much as possible. If you don't mind it when you're playing, that could be a sign that it's not a big deal. In a mix, little squeaks will get lost. But you may also just not be listening closely when you play. Also keep in mind that unless your amp's speaker is at ear level when you play (it's good if it is), you're missing out on a lot of highs that a microphone that is placed right in front of the speaker can hear.

If you can post a sample (wav file with no EQ or effects) I can check it out. Depending on how bad the noise is you may be able to EQ it out, probably around 2.5k range.

You can EQ a lot of it out, you can technique a lot of it out.

May personal favorite studio trick: Just wet your finger tips. That will kill about 90% of the squeek.

If your computer speakers are on while you record with a mic, you may be picking interference up from them as well.

Nothing wrong at all with mic'ing a cab into your computer. Many people (myself included) like that sound better anyeway, and typically, I will blend a mic and direct signal for my recordings.