The key to any finishing or refinishing job is proper preparation.
1) Clean the deck, railings etc well, removing as much dirt, grime etc with a power washer on medium setting
2) Wet the area with a mixture of 2/3 gal water, 1/3 gal bleach, and 1/2 cup TSP.
3) Let the mixture set for 5 minutes, then scrub it in with a stiff bristle push broom wetted in the same cleaning solution.
4) BEFORE the cleaning solution dries, rinse it off with a hose or your power washer.
5) Repeat this in any areas that still look bad or stained with BBQ leftovers etc. This should brighten the wood considerably.
6) Repair any loose boards, nails, screws etc.
7) Let the wood dry completely, usually a couple of days under good conditions before staining.
8) I like to apply stain with a roller on a pole or 1 gal pump sprayer, then immediately over brush with a good china bristle brush to even it out. Don't be surprised if it takes two coats to look good again.
9) Don't wait 10 years again to freshen up your deck!!!!! Good Luck
Site-built deck drainage
Here is an article in Fine-homebuilding on site-built deck drainage.
In this article they use EPDM rubber roofing material.
I wouldn't hesitate to use this or creative alternatives, but I wouldn't consider this as a "waterproof/roof" system. I would consider it rain-proof where it diverts rain and rain-runoff from underneath the deck.
Retail deck drainage
There are products available that are designed for this as well. Like Timbertech Dryspace for example.
My opinion
Using either a site-built or retail product, I still wouldn't store anything under there that would be ruined from getting a little bit of moisture now and then.
If you want a real waterproof roof, then build it as a code-approved roof with code-approved materials installed how they were intended to be installed.
Issues
One potential issue I can think of right off the bat is that dirt and debris will collect in there, so do you have a plan on cleaning this out? You will need access from time to time to at least stick a hose at the highest part to rinse it out.
Another potential issue is that this would limit the airflow and potentially increase the humidity between the decking and drainage material. This could contribute to things like cupping or premature finish failure on wood decking.
Real Roof (info as requested in comments)
60mil vinyl can be considered an approved roofing surface. You would use something like this if you are not putting another surface over the deck.
Another roofing option (if you wanted to add wood planks) would be to do flat roof (i prefer a torch-down roof): then build panels out of your deck surface, or attach the decking to sleepers (see image below), or use wood tiles
Best Answer
Regrading the soil is a must since the soil level was 6" below the rest of the area. The landscape fabric keeps weeds out but not water so I wouldn't be using that. The plastic would have a limited life and you don't want to be ripping up deck planks to check it. I'd be thinking about installing flashing under the deck planks and sloping it down away from the foundation where you think the problem exists. You can get flashing in 50' rolls 20" wide at most home stores. Hope this helps a little. Good luck.