Once the ints have been read in

They're actually generated earlier in the algorithm and will be different from run to run.

how many lookups of them will be performed relative to the number of ints?

That depends upon how long is a (DNA) string. For the whole human genome, circa 3e9 loopkups.

Is there any structure to the ints - e.g. do they cluster together or are they random 64-bit values?

Effectively random and distributed across the full 2^64 range.

The other thought I'm having is to mod the values and use that direct as a hash for inserting into a pre-allocated array -- I can calculate how many values will be generated in advance.

Then the problem becomes how to deal with collisions.

My thinking currently is that presize the array to the next large prime number and the use (hash+prime)%prime to step around the array until an empty slot is found.

For any given starting point that should take me through every single slot before returning to the first, by which time I will have found an empty slot.

What I don't have a feel for is how many collisions I'm going to get as I approach full? And how much bigger do I have to make the array to reduce that to a reasonable number? (I'm coding up a test now.)