A balanced signal has nothing to do with the distortion you were experiancing. The major difference between balanced and unbalanced is that a balanced cable is far less suseptible to interferance and can be run for hundreds of feet, that is not so with an unbalanced cable. The signal is the same.
There are a couple of things that could have caused the distortion on you were getting. The fact that you turned down your bass and it went away indicates that you have an improper gain stage somewhere. Either the gain on the channel you where going into was set too high or the output of the mixer was set too high, either way by decreasing the level at your bass would decrease the level of the whole signal chain.
If I were to be using that mixer I would plug my bass into channel 1, bring the trim up to about 10 o'clock, bring the channel 1 level up to about 12 and then start bringing up the master untill I had good level showing on the leds. The import thing here is to get good level on the channel you are using, in this case 1, then use the master to control how much you are sending out to your computer, if distorts turn it down, if it is not enough turn it up a bit. If you find that not matter what you do you can't get enough signal or its way too much. The trim knob on that channel is the largest gain stage you have on that mixer. Bringing it up or down will alter your level greatly, that is why I suggested starting at 10 o'clock, though depending on the bass you may need to get all the way up to 3. Whatch those LED's, that is what tells you how hot your signal is.
As far as connecting to a PC I would get a dual RCA to 1/8" TRS cable.
HOSA Tech. Then I would attatch the RCA's to the Tape outs on the mixer and then use the 1/8" jack to go into your computer. The reason I say to use the Tape out is because it is an exact copy of what comes out of the main, only on RCA and that leaves your mains available to feed another source that requires 1/4" for input.
Hopefully all this makes sense, you really don't need a DI at all, at this point. It is recomended that you use a DI before you go into a mic pre, such as the example above, but you will be perfectly happy just going straight into the line input on your mixer.
Good luck. Also, when you can afford a better sound card, something with 1/4 or XLR inputs, do it.