Here's a way to do it just from the Dynkin diagram. 
Maybe this only works for Weyl groups, though.

There is, up to scale, a unique subadditive function on the Dynkin diagram, which we can assume is normalized so that its entries are positive integers with gcd $=1$. We can compute this using "Bert Kostant's game" (see these lecture notes for a description of this game https://www.dropbox.com/s/zipwgw3ljkr7sjd/MIT-18-218.pdf?dl=1). 

Let $(a_1,\ldots,a_r)$ be this subadditive function. Note that we have $\theta = a_1\alpha_1+a_2\alpha_2+\cdots+a_r\alpha_r$ where $\theta$ is the highest root and the $\alpha_i$ are the simple roots.

Then the Coxeter number $h$ is equal to $1+\sum_{i=1}^{r}a_i$ and the index of connection $f$ is one plus the number of $a_i$ equal to one.

The order of the Coxeter group is $f\cdot r! \cdot a_1\cdots a_r$, so the previous information is enough to determine the order. See Theorem 13.17 in those notes.