Flooding in back yard, how much drainage do I need.

Hello,
My back yard is about 50' x 160' with the main area of flooding about 20 x 60 in the center with other sporadic spots around the perimeter. The previous homeowner tried to contain the flooding to the center by building small berms in a rectangle to make a hockey rink. He laid plastic down to contain water with boards and all. I was going to use a bobcat to level out the yard the best I could. My yard is kind of natural low spot and the old farm field behind me was tiled years ago(not sure if this is making it worse).I was planning on doing a French drain to fix the problem. My questions are how much drain should I use, and should I bring in some dirt to raise the "hockey rink" up a bit to the rest of the yard?

Thanks!

 

Best I think would be to bring in soil so you can grade the surface to drain naturally. If that's not an option would I consider adding a French drain but I would rather get the water to run off on it's own.