How do I obtain the second museum upgrade?
As this page explains, to unlock the second floor you must have "talked to Blathers on at least 14 different days, donated at least 20 items to the museum, with at least one in each section (insects, fish, paintings, and fossils.) Once the criteria has been met, Blathers will be seen in a thinking pose. Talk to him to unlock the Museum Renovation public works project"
Once you complete that public works project you will get the second floor.