For state letters, there was an official post by this time, Henry VIII had set it up. This would have carried the correspondence between between Elizabeth I and James VI, who exchanged personal letters in this period, as well as official state business. This was through a post roads set-up. So looking into that would probably give you useful information. There's also an official secret service by this period, too.

That said, that would be a courier function with multiple riders, so not what you want if you want a single character to make a journey. But I think I'd probably start with looking at how it was formally done, as in the course of that you'll probably find information about general travel, and maybe people like Elizabeth's major ministers who presumably did send individual messengers on various jobs (earlier, also Cromwell did so).