Nobody loses a deposit over a dusty living room. Deposits go on the specific, checkable things a property manager can photograph, and those things are almost always in the kitchen, the bathrooms, and the edges of rooms.
Start inside the appliances. The oven, the refrigerator, and the microwave get opened at every inspection, and a wiped exterior over a dirty interior is worse than not cleaning them at all. Racks come out, shelves come out, and both get washed separately.
Then go inside every cabinet, drawer, and closet. Empty means empty and clean: no crumbs in a drawer track, no rings on a shelf liner, no dust in the top of a closet. This is slow work and it is the part most people skip when they clean their own unit the night before.
Next, the edges. Baseboards, door frames, switch plates, air vents, and window sills, all hand-wiped rather than dusted. When a home is empty, every one of those lines is visible in a way it never was with furniture in the way.
Bathrooms come after: grout, shower door tracks, faucet bases, and the floor behind the toilet. Hard-water scale is worth treating specifically, since it is one of the few things an inspector will photograph as damage rather than dirt.
Floors last, always, because everything above them sheds while you work. And leave the walls a look: scuffs from furniture move easily with the right sponge, but a scrub that takes the paint off costs more than the mark ever would.
Timing matters as much as the checklist. Book the clean for after the movers are out and before your walkthrough, and tell your cleaner the inspection date rather than just asking for a day. The end of the month fills up first.