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 [][1]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 $q$.


  [1]: https://www.jstor.org/stable/2322289