Step 1: Set the total count.
A chessboard has $64$ unit squares. Choosing any $5$ squares gives $\binom{64}{5}$ equally likely selections, the denominator of the probability.
Step 2: Identify diagonals with enough squares.
A set of $5$ squares can lie on one diagonal only if that diagonal has at least $5$ squares. We look at both diagonal directions.
Step 3: Lengths of the diagonals.
In one direction the diagonal lengths are $1,2,3,4,5,6,7,8,7,6,5,4,3,2,1.$ The same set of lengths occurs in the other direction.
Step 4: Count favourable sets in one direction.
On a diagonal of length $L$, the number of ways to pick $5$ in a line is $\binom{L}{5}.$ Summing over the lengths (only $L\ge5$ contribute): \[ \binom55+2\binom65+2\binom75+2\binom85=1+12+42+56\cdots \] which totals $112$ for one direction.
Step 5: Use both directions.
The two diagonal directions are separate, so multiply by $2$: favourable $=2\times112=224.$
Step 6: Write the probability.
So the probability is $\dfrac{224}{\binom{64}{5}}.$ \[ \boxed{\dfrac{224}{{}^{64}C_5}} \]