A six-sided die is rolled 5 times. The roller wins if the last roll is the same as one of the previous rolls. What is the probability of winning this game?

I tried some but the closest I get was $\frac 16 \cdot \frac 16$, but it's not right.
Write down the results of the five rolls and then read them backwards.  You win if the "first" result is ever repeated, and you lose if it's not.  The probability of losing is $(5/6)^4$ -- i.e., a given roll does not repeat the "first" result with probability $5/6$, and there are $4$ "subsequent" rolls -- so the probability of winning is 

$$1-\left(5\over6\right)^4={1296-625\over1296}={671\over1296}$$
I am not gonna go with massive calculations rather simple logic.

I think the probability will change with each dice roll.

On the first roll it doesn't matter what comes up as one cannot win on first roll(according to the rules you stated).
Now on second roll the probability of winning will be 1/6 as the probability of repeating what came up on first roll is 1/6.
On the third roll however the probability of winning shall increase to 2/6 and so on.