Work on your vocal technique, in particular mic technique. Maybe you're too far away from the mic. Also as mentioned before, check your input gain.
I've worked with plenty of "quiet" singers, and with good mic technique, it's never been a problem.
Stick a decent condenser mic close enough to a buzzing bee and it will get recorded well.
Point being, mics are sensitive and will capture what is there if set up right.
I'd avoid compression at this point because even though it will bring the average level of your vocal up, it will magnify noises like crazy. And that sounds like a major issue you're having.
What you're running into is a signal to noise ratio. The cassette will have a higher noise floor than wider tape formats, or digital formats. But Ive gotten great vocals on 8 track cassette which is twice as narrow as 4 track. Its all in the gain staging going in.
And that has to do with your distance from the mic, your own volume, and pre amplification. One other point. You singing into the right side (front) of the mic?
Once all that stuff is right, THEN consider compression. And with good mic technique sometimes that's not even needed.