Is this a slang term?

No, not that I know of. Can you provide a complete sentence?

Example: I am staying in the same hotel as Gary Barlow. I am fully stalking him.

Thanks

'Fully' and 'stalking' are not a collocation here.

'Fully' is slang for really, totally, completely, a lot, very much so, or to add emphasis to something. Also said as an acknowledgement meaning O.K or I agree/concur. "You should fully quit." "I can't believe Jack is in LongBay! —
Yeah, fully."

Stalk = pursue persistently and, sometimes, attack (a person with

I see what you're getting at. I don't think I've ever heard this particular term before, but I like it (you might have just invented something here); it really transmits what you're trying to say: he's a star, you're a fan, you've managed to get into the same hotel he's at, you might get to see him or talk to him, etc.