While Deligne-Lusztig theory gives a method to compute the irreducible characters of finite simple groups of Lie type, it is not always so easy to find a closed form expression for their number. See, for example, the AMM paper by Benson, Feit and Howe on the behaviour of the generating function for the number of conjugacy classes of ${\rm GL}(n,q)$. Of course for $q >2$, this is not a simple group, but the formula for the number of conjugacy classes of ${\rm PSL}(n,q)$ will be at least as difficult for large $n$ and $q$.

While Deligne-Lusztig theory gives a method to compute the irreducible characters of finite simple groups of Lie type, it is not always so easy to find a closed form expression for their number. See, for example, the AMM paper by Benson, Feit and Howe https://www.jstor.org/stable/2322289 on the behaviour of the generating function for the number of conjugacy classes of ${\rm GL}(n,q)$. Of course for $q >2$, this is not a simple group, but the formula for the number of conjugacy classes of ${\rm PSL}(n,q)$ will be at least as difficult for large $n$ and $q$.