One way of answering this question (although, arguably not the best way) is to look at the author affiliations of papers in bioinformatics journals. I wrote a quick script to tally first author affiliations from several journals where bioinformatics-related articles are often published:

Briefings in bioinformatics, Bioinformatics, PLOS Computational Biology, Nucleic Acids Research, BMC Bioinformatics

And here are the top 20 universities with the most publications in these journals in the last 5 years:

1. University of California
2. Stanford University
3. University of Cambridge
4. University of Oxford
5. University of Toronto
6. University of Washington
7. University of Michigan
8. University of Edinburgh
9. Harvard University
10. Princeton University
11. The Ohio State University
12. University of Illinois at Urbana-Champaign
13. Yale University
14. Johns Hopkins University
15. Nanyang Technological University
16. University of Pittsburgh
17. University of Wisconsin-Madison
18. Boston University
19. Duke University
20. University of Maryland

The academic year has already started in the UK, and this is when most PhD students start. However there are continually positions offered throughout the year.

I would personally recommend Edinburgh, however Imperial, York and Cambridge are excellent. The best site to check in the UK is www.jobs.ac.uk, I found mine through but also ensure you check university recruitment sites

In the UK the funding bodies are MRC, BBSRC EPSRC, Welcome trust, and Cancer Research UK. If you want to work with a specific professor/University and they don't have a role open, they will typically take you if you can fund/source funding yourself.

MIT, University of Washington, Washington University, UCSD, University of Michigan-Ann Arbor, Stanford, CMU are probably good bets.

There is a good phd position in computational genomics in Anderson's Lab (Denmark). The deadline to apply is 31st october 2016. Here is its link http://employment.ku.dk/faculty/?show=857805

Check this Google spreadsheet for many options: https://docs.google.com/spreadsheets/d/1XONt2glhzuzFMcwvb_6PocD9EqkwCBkrP27kb1BCljs/edit#gid=0