Honestly it's not much different from the MPH application process. A lot of schools use SOPHAS for PhD programs as well, and it's the same requirements basically: transcripts, GREs, SOP, 3 LORs, etc.
I'd say the waiting has been the most nerve-wracking part. There's a wait after applying to hear if you get interviews, then a wait after interviews to get a decision, and then a lot more waitlisting than for MPH programs after that. I sent all my applications mid-November and only have decisions from half of them now.
Also, obviously, it's much, much more selective than MPH programs. I've never seen or heard of anyone not getting into any of the MPH programs they applied to (except maybe people who only apply to Harvard or something), but I know multiple people with strong academic and public health backgrounds who have applied and re-applied to PhD programs for multiple years before getting in somewhere. I'm very thankful I haven't had that experience.