How to obtain an upperbound for knots up to k crossings? I think I've found something which involves the genus but I'm not sure.
There are some known exponential bounds on the number. For example, if kn is the number of prime knots with n crossings, then Welsh proved in "On the number of knots and links" (MR1218230) that
2.68 ≤ lim inf (kn)1/n ≤ lim sup (kn)1/n ≤ 13.5.
The upper bound holds if you replace kn by the much larger number ln of prime n-crossing links.
Sundberg and Thistlethwaite ("The rate of growth of the number of prime alternating links and tangles," MR1609591) also found asymptotic bounds on the number an of prime alternating n-crossing links: lim (an)1/n exists and is equal to (101+√21001)/40.
Dowker codes can be used to get an (over)estimate for the number of knots with $k$ crossings. Hoste has written a few, extremely clear, papers on using Dowker codes for enumeration of knot tables. I don't see how genus could be used - crossing number is an invariant defined in terms of diagrams while genus is much more topological... Very curious!