In threads about cleaning shower stalls, several people have recommended Easy-Off Oven Cleaner, fume free, blue can, as a great tile cleaner.

The Easy Off is generally suggested for fiberglass tubs, and you need to leave it on for a couple of hours. For ceramic tile, I would go with a good tile cleaner. I like the Stone Tech products, but there are others out there that are also good.
Stone Tech KlenzAll

409 and a scrub brush...or maybe comet! LOL!
It looks like "just dirt" to me....when I moved into this house, a beautiful quarry tile floor had many MANY coats of self polishing floor wax...like Mop and Glow..on it. I covered it with paper toweling, several thicknesses, soaked it with 409...topped it with saran....and let it sit for most of the day. Scrubbed off the loosened stuff and set another area to soak over night and cleaned that in the morning. Took most of a week, but It got done!
Your mess doesn't look that bad.

I'd get some Simple Green and a scrub brush and a LOT of paper towels.... and maybe some knee pads.....I also think you might need a grout cleaning brush, they work great on grout. Please keep us update, interesting job you have there....

If you have access to the bathroom cleaner Scrubbing Bubbles, I would try that. I haven't found anything that works better on tile- made my dingy, greasy grouted kitchen tile look almost new. I've never tried Easy-Off though, not brave enough.

Bathroom cleaners are usually low PH and notably hard on grout. I wouldn't us it on anything but porcelain or maybe fiberglass....but nothing with grout.

A scrub brush and dawn dishwashing soap and a bucket of water. Or maybe Tide detergent and water. It looks like there is years of dirt built up there, so you'll need to go over it several times. As a former landlord, I sympathize. Hard to imagine people could live with such filth, but as a landlord nothing surprises me anymore, I've seen a lot worse. By the way, what should one use to clean bathroom tiles? We recently renovated our bathroom and I've been using scrubbing bubbles to clean it. I don't want to ruin the new grout, what should I use instead?

Biz laundry stain remover and vinegar is great for this. Biz comes in a big box like oxy clean and it is cheap. My local walmart doesn't carry it but the target did. Spread it all over in small sections and pour or spray vinegar on into a bubbling paste and leave for a couple minutes. Scrub away with a hard brush and your stains will literally lift away. You may need to spot treat certain areas if they are resistant. If you don't rinse with water well when you are finished it leaves a powder that dries. I usually try to sweep that up and do a final mop with vinegar water. Also Biz is just an amazing cleaner in general. I use it on my stubborn soap scum in the tub as well.

After you finish cleaning that grout, buy some waterproof grout sealer at a tile store and brush it onto the grout lines with a little foam brush. Not necessary to seal porcelain tile.