Imagine tossing the dice one at a time, and recording the results, or equivalently labelling the dice A to E, and recording the results as a string of length $5$, result on A, result on B, and so on. There are $6^5$ possibilities, all equally likely. Now we count the favourables.
The "straight" part can be of any of type $1,2,3,4$, $2,3,4,5$, or $3,4,5,6$.
First we deal with the $2,3,4,5$. In order not to get $5$ in a row, we must avoid $1$ and $6$, so we must double up something. What we double up can be chosen in $4$ ways. For each of these ways, the smallest non-doubled number can be placed in $5$ ways, then the second smallest in $4$ ways, then the third smallest in $3$ ways. Now the doubleton falls in the remaining spaces. That gives a total of $(4)(5)(4)(3)=240$ ways to have $2,3,4,5$ as the "straight" part.
Now we deal with $1,2,3,4$. We could have the other number be a $6$, and then the $5$ numbers can be arranged in $5!=120$ ways.
Or else we could double up one of our numbers. We have already analyzed this, and seen there are $240$ ways to do it. So there are $360$ patterns where the straight part is $1,2,3,4$.
Similarly, there are $360$ patterns where the straight part is $3,4,5,6$.
So our total is $240+360+360=960$, and the required probability is $\dfrac{960}{6^5}.$