First would be to eliminate the low spots that allow the water to pool, add fill to bring it almost up to your clearance minimum.
I would then Slope the grade under the deck slightly from the middle to all sides. Then dig a french drain around the perimeter just inside the rim joist and sloped so it all flows to the back side and then continue it out and away to the trees.
This is not my forte so other may have better ideas or point out the flaws in mine.
On a structural note: I am worried about the lack of beams. I would like to see your joists sitting on, or anchored to, beams and those beams on pier blocks. It is floating deck and it will move with ground fluctuations. As built i foresee that movement weakening or loosening your structural integrity.