Gloves. Trash bags. Sponges you plan to throw away. I’m going to advocate bleach here (diluted, of course, as bleach should always be). Good ventilation (open the windows and get a fan going if you can). Fill the tub and sinks with hot water. Add bleach. Add bleach to the toilet. Let that all sit for a while. Start in the worst room (kitchen or bathroom). Methodically get to every single surface. Switch out your sponges or rags frequently. Take breaks that include fresh air frequently.
You could always put the pressure on your landlord to have the place cleaned, but: A) I’ve lived in the real world and dealt with a lot of landlords; B) that takes time, time when you’re still in that filthy space; C) this way, you know it’s being cleaned to your specifications.