It has been a while since I've done any cyanotypes, but I used to do a lot of it. I don't think a lot has changed in those few years, the process now being something like 160 years old.
I suspect that your paper has no sizing. You CAN use unsized paper; I did that all the time, but I used Arches 88, which is a very expensive and beautiful paper made for intaglio printing; the cyanotype soaks in just like blotter paper, and that introduces lots of interesting problems but makes it possible to get great "black". By computer paper, don't you mean ordinary white paper like "copy paper"? Sizing using starch or arrowroot might help a lot, but as someone above suggested, why not just get some watercolor paper? It comes presized, and is much better paper with much heavier weight; it will hold up much better in the wash and since the cyanotype doesn't soak deeply into the fibers it doesn't require the longer wash times.
The suggestions above regarding color changes indicating exposure are right on.
In sunlight here in western Washington (maritime climate) an exposure of 1/2 hour would be about the minimum even on our unfortunately somewhat rare clear days. Despite the obvious climatic differences, it would probably not be a lot different where you are, since the sun is still 93 million miles away, but you may be getting a higher proportion of UV which would shorten it a bit; I don't know, but maybe somebody does. A rather heavy, fairly contrasty negative without a lot of base+fog would be ideal.
Do consider making photograms, just laying stuff on the paper using no negative at all. It's a really great medium for that. Three dimensional objects produce varying degrees of value and sharpness/blur, and transparent objects introduce very interesting images because they become lenses of a sort, concentrating light in some places and dispersing it in others.