have you tried hot water and bleach, on your hands & knees scrubbing in sections?

I would first try putting baking soda or kitty litter on the stain to absorb some of the grease. You will have to leave it on for at least 24 hours. Then use a degreaser. Good luck.

Thanks for your advice!

I will try both and see how it goes.

Can you give me an example of a degreaser please?

This is a great cleaner/degreaser I found at Ace Hardware a few years ago. Works great on grease stains, and even on laundry stains. It doesn't smell great; just works great. I'm sure other stores sell it as well. Greased Lightening.
http://www.glblast.com/products/Products.aspx?ProductId=1

I am living in Montreal so I am not sure if my local Canadian Tire will carry that "Greased lightning" product though I am sure they will have something comparable. If that doesnt work, then I will do the carpet thing.

I know this will sound nuts but when I am on my deck reading a book, I get distracted by the grease/dirt and just can't relax. I really need to give it a good cleaning....I wish I was one of those people who just didn't care. : 0 )

That is a good idea as well. I will look in to all these options.

I am a teacher and this is my last week of school and then I will have time off to do some things around the apartment (i.e. clean the deck, wash duvet cover, clean the moldings etc..) I am looking foward to catching up on some housekeeping.

Wait till late at night, make sure your neighbors aren't around. Clean your deck with water. It will dry by morning and no one will be offended.

Try some of the degreasers used for cleaning car engines.
Auto dept. Canadian Tire. Then put down a mat...

I get Greased Lightening at the local hardware store too. They have a huge section of household cleaners.

Omg... Lauren I thought it was just me some even say I am way to fussy about this. I also share the same problem u have with my second floor balcony. The previous people left it in such a mess. Ive tried the typical cleaning with household products. It does not work I must say. I need something much more stronger like the Greased Lightning mentioned. I wanna try it. I hope it works!!. Thx Lauren and everyone for posting about this this has helped me tons.