There is, in the episode, no proof that Kirk did anything more than some making out and drinking.
It is likely, given the character, that the innuendo following is accurate in it's implications... but it's not clear.
(Kirk is alone in the room, and is startled when a curtain is pulled back and the blonde girl appears, wearing very little.)
DRUSILLA: I was told to wait for you and provide wine, food, whatever you wish. I am Proconsul's slave Drusilla. Although for this evening. (pause) For this evening I was told I am your slave. Command me.
KIRK: It won't work.
DRUSILLA: What will not work?
KIRK: Whatever he has in mind, whatever tricks. You hear that, Proconsul? It won't work. I'm not co-operating. I may die, but you won't get any entertainment out of it.
DRUSILLA: We're alone. Please believe me. I've never lied to one who owns me.
and after a scene with Kirk and Spock...
(The wine is being poured.)
KIRK: Very good. (he tries a grape) Excellent.
DRUSILLA: I was concerned. I am ordered to please you.
KIRK: I've been in some strange worlds with strange customs. Perhaps this is considered torture here.
DRUSILLA: Torture? I do not understand. I do not wish to see you tortured in any way. (she kisses him) At the first sign of pain, you will tell me?
KIRK: You'll be the first to know.
(Later, when the oil lamp has burnt out, and Kirk is asleep)
CLAUDIUS: Captain. (Kirk wakes with a start) I'm sorry I was detained. Should we have our little talk now? So far on this planet we've kept you rather busy. I don't wonder you slept through the afternoon. By the way, one of the communicators we took from you is missing. Was it my pretty Drusilla by any chance?
(Merik enters. There are guards in the room now too.)
CLAUDIUS: See if he has it. Not that I would have punished her. I would have blamed you. You're a Roman, Kirk, or you should have been. It's not on his person?
MERIK: No, Proconsul.
CLAUDIUS: I am sorry I was detained. I trust there was nothing further you required?
KIRK: Nothing except perhaps an explanation.
CLAUDIUS: Because you're a man, I owe you that. You must die shortly, and because you are a man (pauses) Would you leave us, Merik? The thoughts of one man to another cannot possibly interest you.
CLAUDIUS: Because you are a man, I gave you some last hours as a man.
Claudius is implying, not stating, that Kirk had relations, proving his Roman Maleness. Note that Roman concepts of Maleness involve being the penetrator - either sexually or martially, a Roman "Man" is one who does the penetrating. So, with a bit of historical knowledge, Claudius at least thinks Kirk has had sexual congress of some kind with her.
But we also do not see her confirm it, nor do we see her nor Kirk en deshabile... So if the did the deed, he got dressed after. Also, note that Drusilla's outfit is extremely easy access. Kirk's isn't so easy.
Claudius also implies that he thinks Kirk used her to steal a communicator; we later find out it's with Merik...
This, coupled with Drusilla's statement, very nearly confirms that Claudius didn't monitor them. So Kirk might not have gone all the way, but he certainly would have enjoyed her physical presence once he realized she wasn't a trap. But it's far from clear that he actually did anything besides kiss her.
It's also clear he's letting her "lead"... So if anything, he's avoiding "Taking advantage" by letting her be the active partner.