If the previous episode was where Hartnell started to shine, "Desperate Measures" is where he makes it absolutely clear that the series belongs permanently to the Doctor in general and to him in particular. It's not that any of the other regulars become any less interesting--Ian and Barbara get some great moments together, and Vicki has an instant rapport with them. The scene where they try to explain their status as unwilling time travelers, and Vicki's response ("You're five hundred fifty years old?") and Ian and Barbara's response to Vicki's response...it's all wonderful stuff.
But the Doctor becomes far more significant. There are two key scenes where he demonstrates his absolute mastery of the role, and in the process centers the narrative entirely around himself. The first scene is his initial meeting with Vicki; it comes at a key point in the narrative for her. The rescue ship is due in perhaps a day, and she'd been hearing for years that her only chance of survival was to listen to Bennett and Koquillion and wait for help to arrive. Ian, Barbara and the Doctor are strangers who immediately disrupt her world on every level--they mock her tormentor, they have a risky plan with very few upsides and potentially catastrophic downsides, they're potentially antagonizing Bennett (it's never overtly stated, but the way she reacts to that tells you volumes about the way he's treated her all these years) and to top it off, Barbara just killed her pet. She has absolutely no reason, no reason at all to trust any of them.
And then Hartnell turns on the charm. His eyes twinkle, he smiles a warm and brilliantly paternal smile, he sits down and beckons her closer, and Vicki just melts--and the audience melts right along with her. This is entirely a new experience for us--Hartnell's had his moments of vulnerability and warmth, sure, but we have never been subjected to anything remotely resembling a William Hartnell charm offensive. For someone following the series since 'An Unearthly Child', who remembers his initial scenes with Ian and Barbara, this is scarcely even credible. He is transcendent here, bringing a degree of warmth and kindness to this scene that makes Vicki's decision to travel with him instantly believable.
The second scene, of course, is his confrontation with Bennett. It's a magnificent piece of work, from his initial scene sitting in the chair and talking with his back to the villain of the story all the way up through the confession and his snarled, righteously furious, "You're a madman!" He owns this scene, every second of it. Bennett is great too, don't get me wrong; Ray Barrett hits every note of his character perfectly. But it's that very perfection that gives Hartnell the chance to take over the part so completely; Barrett as Bennett is bringing forth every ounce of his charisma and menace, and Hartnell is matching him with an implacable determination that gives him instant authority and gravitas. Obviously, it'll be a while before he can bring that authority to the physical arena--in this case, he's saved by Dido ex machina and a convenient cliff. But he's unquestionably asserting himself as the moral and narrative center of the show. From this point on, Ian and Barbara's days are just as numbered as Susan's were--this show now belongs in spirit as well as name to the Doctor.