Is installing an egress window in a basement a do it yourself job?
Before going through the trouble of an egress window, make sure the water problem is tackled. That, itself, can be a huge chore.

And egress window CAN be DIY, but it's going to be a lot of messy work. You need to move a large chunk of soil, cut concrete, install a well, and given the water issues, likely a dry well or drain system of some sort. 

But yes, it can be DIY. I would worry about bowed walls, however. If the foundation walls are bowed inward, that can be a sign of huge water issues, as well as structurally compromised foundation walls. Before you dig around the foundation, you may want to consult with a structural engineer first.