The key to finding the "last day of a month" is to realize that the "last day" of a month is always one day before the "first day" of the next month. As the date of the first day of a month is easy to construct, you just have to construct the first day of the current month, then go to the last day of the month before. Then extract the year and month of that month, and construct its first day from it. Repeat until you have skipped enough months.
Personally, I like to use Time::Local and localtime, together with POSIX::strftime to manipulate and create date strings. The more heavyweight and somewhat different approach is to use DateTime, but as I often have to do date calculations, and not always in Perl, the strftime approach has worked more often for me than the DateTime approach.