That's my cat!!!!! Molly. LOL It looks just like her - - but she's sitting in my lap - so I doubt it really is.
So, here's MY advice:
1) you can put up flyers in the area to see if someone lost a kitty.
2) you need to decide if you are going to keep the kitty or not. If you are going to keep the kitty, you need the following items:
Litter box, scoop, kitty food (NO FISH FLAVORS), litter, and kitty toys, and a brush.
The kitty will come to you when it wants attention - - pet it - and let it go back to being lazy.
If you keep the kitty - it MUST BE spayed/neutered. That is an ABSOLUTE MUST.
It also needs shots, possibly deworming, etc.
This can get expensive. I would call a rescue and see if they can work with you to get this done for a cheap price. Maybe they will help you out.
Where do you live? Your stairs look like mine - - I live in NYC.
Editing to add a pic of Molly: