reading of 1 nucleotide takes 1 hour on the fastest machines. You will need to read more than one nucleotide :-)
Could a processor such as an ASIC be customized to do this faster?
1. well, ok, ok, there are single molecule sequencing techniques that do a single nucleotide in miliseconds :-) [Pacific Biosciences] but these are not reliable (20% error rate).
2. ASICs are already used in some sequencing techniques: [http://www.iontorrent.com/] in fact this is a very interesting approach and quite a promissing technology but still not mature enough to compete with illumina in terms of nucleotides / $ ((( but if you can create an ASIC that measures pH changes in wholes of ~50nm diameter ... this would be interesting ,+ it has to have millions of these wholes on the chip )))
........ but in any case there is no connection between biology and bitcoin mining :-)