Once the senate had passed whatever resolution was formally required, what did the process look like?
Was there some sort of investiture ceremony? What sort of religious officials would be involved?
Did they believe that the process itself elevated the dead emperor, or was it considered an acknowledgement of a pre-existing divinity?
I'm mostly asking about the principate in general. Julio-Claudians, Flavians, Antonines, or whatever, though if it changed from one period to the next I'd be interested in learning about that too.
Hi, I answered more or less the same question (I think) a year ago, which you can find here. As for what religious officials would be involved... To get a full answer I'd suggest you pick up Étienne's Le culte impérial dans la péninsule ibérique d'Auguste à Dioclétien (it's in French, but what you like to access in it are the tables of Roman officials). However, more briefly what you want to know is that there were mainly two priesthoods accociated with the Imperial Cult.
The sacerdotes were a sort of administrative priesthood for the worship of the genius and numen of the Emperor while he was alive, as well as the rest of the imperial family.
As suggested by Fishwick, there is a clear change in the priesthood when the emperor dies and subsequently is deified (if he is granted that honour). If that happens, the priesthood accociated with the cult is instead the flamines (the sacerdotes would still take care of the cult surrounding the imperial family, as well as the succeeding emperor). The flamines is a priesthood that we otherwise see for gods, most famously perhaps the flamen Dialis (Jupiter).
I realise that the linked post doesn't answer the question about changes over time in the cult, and if you're interested I can give it a go in answering that as well, but sadly it will involve a decent amount of speculation, simply because our source material for the cult is a lot more quanitative during the 1st century AD, than for the 2nd and 3rd centuries.