How did the U.S. Treasury determine which president went on each different dollar bill denomination?

I don't have a solid answer for you, but I do have a few bits of info that might help anyone researching into this.

Currently, the US Bureau of Engraving (BEP)( is in charge of the entire process. They have several designers that sketch out different ideas for bank notes, but the Secretary of the Treasury has final say. As far as I can tell, Congress doesn't have much (if any) direct influence on the selection process. I haven't found any information indicating that a different process was used any time in the past, but I haven't done much research in that direction.

The original selection process for US bank notes happened in the early 1860's when the federal government started printing money. At that time, the $1 bill had a portrait of Salmon P. Chase (then the Secretary of the Treasury).

In 1869 the portrait was changed to George Washington. The portrait selection has actually changed several times over the years - you can see all the major design changes here.

I don't have access to it currently, but it looks like the book

A Guide Book of United States Paper Money (Whitman)

might be a good source for further reading.

You can also read more about the history of the BEP here (warning, PDF)