Bogazkale is off the Ankara - Samsun road from the town of Sungurlu and there are plenty of buses that run on that route. Dolmuses run between Sungurlu and Bogazkale but the services are unreliable. Locals in Bogazkale often don’t know when the bus arrives or leaves from Bogazkale. Taxis are available between Sungurlu and Bogazkale.
Do note that as of at least 2019, according to multiple sources (tourists and locals), there are no public transport services between Sungurlu and Bogazkale during the weekends. The times which the dolmus runs were quoted as 6:00 and 7:00 in the morning, and 17:00 in the afternoon (Sungurlu), only on weekdays, for a price of 10 TL. The fastest bus from Ankara takes 2 hours to reach Sungurlu, however, this does depend on the route. Metrobus would appear to have the last and first direct services to Sungurlu (23:58 & 5:00), however, if you wish to take the dolmus, it is advisable to take the midnight bus and arrive early into Sungurlu (the Otagar is a 24/7 rest stop for coaches with food), as otherwise you are likely to get stuck (Bogazkale is 30km away). Hitchhiking is a common option, especially as there is only one road from Bogazkale (that only serves 1,300 person strong Bogazkale), which forms an intersection to the Ankara-Samsun highway. Asking to get dropped off at said intersection, then hitchhiking from again from there down the road has been a successful tactic, especially if you look like a tourist. Some, of course, will just be going all the way to Sungurlu, and vice versa. Leaving Bogazkale via hitchhiking after 17:00 can be difficult, but is not impossible.
Taxis from Bogazkale can mark up outrageously for the 30 minute trip to Sungurlu (one was quoted at 130 TL), however there is a friendly, if very insistent driver who quotes 60TL, and can be haggled down to 50TL if necessary, who roams the two major ancient sites in an older taxi.
See photos of Hattusa at Bogazkale