Is there any way in Google Analytics to track whether somebody who completes Goal 2 has, on a previous visit, completed Goal 1?
It looks like custom segments allows you to see people who have completed them in the same session, but not if they’re completed on separate visits.
Reading your question, I think a possible solution could be:
When your user converts on Goal 1, then set a Custom Variable with opt_scope 3 to tag that visitor as having completed Goal 1, say as “Goal 1 Visitor”.
Then set up Goal 2 as normal and in the Visitors>Custom Variables report, you should be able to see the number of times that Goal 2 has been completed by a “Goal 1 Visitor”.