The check comes when the DM determines what the player describes has an uncertain outcome. That necessarily means the players must offer a goal and approach first. How that goal and approach is communicated can be a straightforward statement of what you're doing and what you hope to accomplish. Or it can be acted out (as you say).
In my view, whether you act it out or whether you plainly state the goal and approach should have the same uncertainty and DC, assuming the goal and approach boils down to the same in either way of communicating it. In other words, your acting ability or way with words obviously adds to the play experience, but otherwise has no effect on the mechanics.
The DM then narrates the result of the adventurers' actions. I would add a further limitation on the DM here in that he or she cannot in the narration of the outcome state what the character is doing outside of what the player has already established.
Typically, when I need to resolve an outcome on a player's stated action, I call for an ability check. Then the players decide which of their skill proficiencies, if any, apply.
Pretty much what he said. Especially that 1st line.
I'm fine with either the player speaking in 1st person or just telling me what they aim to accomplish. Once either is done I'll decide if any dice need rolling & the DC.
Here's a humorous tale of tragic success from our PF session several weeks ago on this very topic:
The PCs had successfully infiltrated the enemies fortress by disguising themselves as new recruits. In their exploration they came to realize that they'd missed a magical "key" needed to access a teleporter arch. Without this they were stuck on the current floor. Eveyone remembered me describing the key in a previous encounter (now a 2hr hike away), but nobody thought it important enough to add to the loot sheet.... Or thought someone else had written it down....
Theyconcluded that they had left it behind with the other mundane gear & bodies.
So they spent
at leasta 1/2hour debating what to do. (we're not concerned with things bogging play down)
They concluded that they would need to bluff the guards in one particular room. They decided to spin a tale of being stupid newbs & get the sgt on duty to open the arch so they could get to the mess hall.
OK. As it stood I'd decided upon a fairly simple bluff DC while they planned & re-planned. And what would happen if they failed. The only question was who was going to be making the bluff check as none of their characters have any ranks invested in this skill.... They chose the Paladin - because highest modifiers.
All he had to do was roll the dice & tell me the #. Odds were pretty good that they'd get to the mess hall lv.
But before I could say anything, he declares that he throws open the door to the guard room & shouts in a panicked voice that the wall captain needs reinforcements right now! "You, you, you, go. Now." And says that he points to the first three guards he sees (indicating wich minis on the map. 2 were identicle, 1 different. You know, sgt & 2 grunts.).
And then he rolls the dice. A 19+mods. An outstanding success!
Except that what he said bore NO semblance to the plan he & the other players had spent 30+minutes hashing out.
So, slightly confused, the sgt & two grunts headed off to see the wall capt. This will take about 5 minutes round trip.
The player has his paladin rush into the room & start searching random foot lockers. Completely ignoring 2 facts:
1) That the previous key they remember was found on the sgt of the squad they'd ambushed. They know this fact. And that I use the same mini to denote the sgts. of this force. Another known fact. Wanna guess who had this key?
2) That the room he's entered is L shaped....
The other players? They were like "WTH? What part of that plan didn't you (paladin) understand??"
From around the corner: "What's going on? Hey! What the @#$ are you doing in my foot locker?"
There were 3 more guards. Wich should be no surprise as every other guard squad they've encountered #s 6 soldiers.
Roll initiative. And cue gigantic battle as the pcs switch from trying to get to the mess hall to just trying to escape the fortress alive. This ultimately ends with 3/4 of the PCs down & the war-mage surrendering to the bad guys.
The original plan would have almost certainly worked. All Sean had to do was roll the dice....
But he added the RP & I'm not going to negate that. Nor will I reward you for actually saying the exact wrong thing.