An excellent reference for this kind of descriptions (and for many other facts about $E_8$!) is Skip Garibaldi's paper "$E_8$, the most exceptional algebraic group" in the Bulletin of the AMS [(here)][1].
In particular, in section 4, he describes various possible branchings, and Example 4.4 deals with the branching $A_8 \subset E_8$.

The specific example is rather short, but he includes some further references that might be helpful.


  [1]: http://www.ams.org/journals/bull/2016-53-04/S0273-0979-2016-01540-0/