Meaning & Origin of phrase "Step into [him/her]"

To step into someone is to take a step uncomfortably close to them. It is frequently encountered in sports as a strategy to limit an opponent's freedom of motion or to recover initiative from an opponent with longer reach.

Zezel, a former pro soccer player, would often tie up his man's stick, step into him and kick the puck back to his winger. This is the type of play the creative center must learn to master if he is going to consistently control the faceoff. - hockeyshot.com, "How to Win Faceoffs"

You have options when you’re being defended aggressively in the post. The defender will often work to move you away from your comfort zone on the low block. Try any of the following strategies to create a seal down low:
. . . Face the defense: Turn and face your defensive man and step into him with one foot between his two feet. Turn and pivot, putting your butt into him to seal him off. - basketball.isport.com, "Effective Low Post Moves in Basketball"

The coach said to me, 'listen, keep up the pressure on this young boy. This boy is strong and he could move. So, you have to cut the ring off on him and step into him, and that is what I did. - Jamaica Observer, "Spencer too strong for overmatched Bowen"

Deakins is advising the detective to "get up close and personal" with the suspect, to press him closely with his questioning and give him no opportunity to put himself at an emotional or discursive "distance" from the detective's attack.

It's feasible the screenwriter is faithfully reporting a localised dialectal usage. But I think it more likely he's either ignorant, or simply trying to add interest to the script by using a mangled (but still understandable) variant on established idiomatic usages. Some results from Google...

You should step into him. (1 result, apparently from a non-native speaker)
You should step on him. (4470)
You should lay into him. (7460)
You should lean on him. (20800)

I'll step into him. (1 result, a "literal" usage where the speaker walks into his boisterous dog)
I'll step on him. (27900)
I'll lay into him. (19100)
I'll lean on him. (170000)

I think lean on him is probably the most common term for OP's context. This can also mean get support from him, but here the sense would be harass him / put pressure on him.