I did this, and learned some stuff along the way.
I took an ADO bus from TAPO in Mexico City to Central de Autobuses de Pachuca. TAPO is one of several bus stations in the city, and seems to be the closest one to the airport. They had busses leaving for Pachuca every hour. There were also more frequent busses from the north bus station, seeming to leave every 15 min, but I was coming from the airport.
The ride was just under 2 hours, and made a few stops in the outskirts of Pachuca before finally arriving at the central bus station there. The ticket cost MX$116 at the counter, or MX$82 online at least a day in advance.
I figured there would be a shuttle bus that went up to the El Chico visitors' center, but I couldn't find any info about one online, and nobody that I asked knew. So I took a taxi. The driver wasn't familiar with the park, so I showed him the route on google maps and ended up spending MX$400, plus a MX$100 tip.
Once I got to the visitors center, I asked if there was a shuttle. There is, and it runs hourly back to the town. The last one left at 6:11pm. I think it cost MX$30 🤦
There was no posted schedule anywhere; it seems like you just had to know, or at least know where to ask. The van dropped us off in front of a clothing store just north of the center of town. I didn't see any signage indicating what routes ran from there, but there were tons of vans there, and I bet that's where I could've caught one on the way out of town too if I had known.
I hope this helps someone else out!