Do I need to apply for a visa before traveling to Thailand for 5 weeks as an American citizen?

Do you HAVE to? No. But I would suggest you do.

You can enter visa exempt as an American citizen, and get stamped into Thailand for 30 days with no visa. But if you want to stay 5 weeks you're going to have to go to the immigration office at some point during your trip and get the 30 day extension, potentially spending a day of your time and the money to travel to and from the immigration office and gather all the required paperwork.

If you apply for a Tourist visa through the evisa system before you leave, you can gather all the documents they ask for in the comfort of your home and then arrive in Thailand and get stamped in for 60 days and not have to worry about anything immigration related during your trip.