What is the rule for repeating moves?

I know, that I can claim a remis, if me and my opponent are doing the same move 3 times in a row. But my case was a little different: 

I was about to win the game, but had a little less time on my clock. So my opponent put me in check with his queen from several positions, while I was moving out of check with my king around my own castle.

It was obvious, that his moves could not lead to success and he was just playing on time. 

Is there any rule for this situation. Or is there a kind of gentlemens agreement in check, that you change to other moves, if repeating or similar moves are senseless?

50 move rule.

50 moves without a pawn move or any capture, and you can claim a draw

So there is no kind of gentlemen´s agreement or other rule. I don´t like the idea, that my oppenent can force me into a remis or even a loss due to my time´s up even though he´s way behind. 

Yes, but that's the function of time in a timed game.  Even if it was untimed, your opponent may have been able to force a draw by perpetual check, no?

Next time, move faster.

If you play with a significant increment (such as a 2 | 12 game) you won't run out of time due to your opponent's delaying attempts.

There is no rule on "repeating" moves.

rooperi is correct when he mentioned the 50-move rule.  After 50 moves, without a pawn move of any kind, or a piece captture of any kind, either player can claim a draw.

There IS a rule on "repeating" position.  It's called three-fold repetition.

If the player on move can claim, with his very next move, that the following position will have occured three times, (not necessarily in succession), with the same player to move each time, that player may claim a draw.

Your opponent did nothing wrong.  Winning on time is a big part of blitz chess, which is what you were playing, I assume.  Berder's suggestion is a good one if you don't like losing on time... play with a large increment.

I hope that helps.