For an air purifier, you need one with a UV light.
1. Air filters might reduce the airborne spores, but if you have mold already growing behind some wardrobe you are wasting time.
2. Hydrogen peroxide, chloride, and isopropyl alcohol are all good tools to fight it when you find a place where it grows, but ultimately you have to solve the problem at a deeper level as mold has excellent survival skills (you might clean the wall but the spores in the air infect it again right afterward).
3. In order to eliminate mold, you have to change the environmental conditions. As most living organisms mold requires food and water. The water comes either from inside the wall (that is something only your landlord can fix) or from the air. So try to maintain a low humidity level within your apartment (<50% should be okay). The easiest way to achieve that is by using your heating and replacing all air within your apartment by opening all windows for 5 minutes (to let in cold air, no longer, as it will cool down the temperature of the wall otherwise), three times a day. Mold nourishes itself from organic material. Very often that means fabric, wood or wallpapers (yes removing your wallpaper might be a good idea). Since water condenses in cold spots first, those are the place you want to keep looking for (walls next to windows, corners of a room).
Three years ago I had a mold problem in my apartment. Following the above advice, the mold didn't come back (except for a few spots in the shower, but that is kinda difficult to keep dry).
If you smell "swimming pool" where no pool exists, it might be prudent to leave the area immediately and take a 1000 mg vitamin C tablet, before making any attempts to discover the source of the smell. (Taking vitamin C after the damage is already done won't help.) If the odor is especially strong, or if it causes any irritation to your nose and eyes, evacuate, and do not return. Call your emergency services number and tell them you smelled a strong chlorine odor.
Don't combine bleach cleaners with ammonia cleaners, ever. Don't even store them in the same place. Also avoid mixing bleach with acids, such as vinegar.