The black marks are getting bigger, the grease is building up, and you've been putting off cleaning for a quite a while, but all is not lost, we've found through experience some ways to help reclaim the former glory the grill had when it was new. Also we would recommend you use rubber gloves of some kind.
One of the best ways to cut grease, both old and new, is 409. Spraying that and wiping the grill down with some paper towels will remove all of the newer mess and some of the old burnt in mess. There's not much to it, just thorough wiping and scrubbing all over the grill, inside and out. This is going to be the bulk of the work area wise, but as far as time consumption, the next task will probably take the most amount of time.
The burnt in stains will be the hardest to get out, but it is doable to a certain degree. Using something with a light abrasive like bar keeper's friend will help scrub out the dark marks. Make sure to scrub in the direction of the grain or there will be some visible scratches. Although it will never be as perfect as it was when you got the grill, it will be noticeably better, especially after the last step.
You can also clean the grilling grates with Easy-Off oven cleaner. Allow the cleaner to sit for 15-20 minutes then hose them off with the garden hose. You may do the same thing for the grill itself, after you remove the burners (which you do not want to get wet). The Easy-Off treatment will bring them as close as possible to new.
After you are done with the cleaning, be sure to use an oil based stainless steel cleaner both on the inside and outside to remove little marks and make small scratches hard to see. Most importantly though, it will make everywhere you use it shinier and better looking. After all this, your grill should be looking much better than it used to.
1) 409 or some other grease cutter and wipe it down with paper towels
2) bar keeper's friend or some other light abrasive to remove the burnt in stains
3) easy off oven cleaner on the grates and the dirtier portions of the grill
4) stainless steel cleaner to polish and make shiny