There are 3 basic cases, and then random stuff on top of that:
1. Your civ is healthy with a mountainhome with a monarch. The monarch will basically only arrive if summoned or the mountainhome is destroyed.
2. Your civ is dead. This means no dwarven caravans or liaison will ever come, nor will migrants past the two first waves. No monarch will ever be announced, although it is possible it might be possible to get one after creating and retiring multiple fortresses. It's rather difficult to get a dead civ, and there is no known way to ensure you'll get one: The civ can have lost its only site in the first year and no dorf has lived for 1000 years, and there's STILL a void sending caravans, so the civ is actually not quite dead, somehow. There's a warning message on embark that's common for case 2 and 3.
3. Your civ is struggling. it may or may not have a monarch, but it has no sites (not counting caves and camps here), or at least none of note. If/when the monarch bites it, there's a very high chance one of your dorfs takes up the mantle. If your civ doesn't have a monarch on embark, some unknown factor may trigger the rapid or somewhat later "after a polite discussion...." message, but if that doesn't happen, you'll get a monarch at the turn of the year (if I recall correctly, the second turn of the year, but it might be the first), and with a different coronation message. It seems you can influence the monarch you get by appointing the one you want to have as the expedition leader (not 100%, but it seems to increase the chances).
The only known way to distinguish between cases 2 and 3 is to embark and bring up the 'c'iv screen: if it's completely blank the civ is dead. If there's a dwarven civ there it's merely struggling (or healthy, but the the presence/absence of an embark warning will mark that difference earlier). Expanding the civ you can see if there's a monarch in the list.