How the primogen council is chosen can vary from city to city, but there are a few standard themes to the process.
This method is fairly uncommon in the system, and requires a Prince who is despotic and powerful enough to choose primogen even against the will of their clans. Such appointed primogen rarely have influence over the prince's decisions, mainly because the Prince put them there herself. In other instances, a primogen can be appointed from within the clan, a resigning primogen can name a successor, or a clan elder (such as in strongly heirarchal clans like the Tremere) can name the person who sits on the counsel.
One the most common means of selecting a primogen, clan vote on who can represent them best. Though it seems very democratic, the process may not be as straightforward as that. Boons can be called in to sway the votes and sometimes the most powerful or most reputable clan member is not chosen due to personal grudges from other clanmates. It is possible in the case of the "election" method that the lowest denominator is chosen, because the clan itself ends up choosing the one person whom no one seems to have a grudge against or who hasn't made waves in the past - the lesser of all evils
Similar to the election process, this primogen gains his seat due to clan popularity. He is seen as a person who is helpful and gets the job done, and many clan members are willing to follow his lead. Usually primogen who gain position by popular acclaim are the most effective, since they were not appointed and not voted in (as the lesser of evils) and truly seem to have the clan's best interests in mind. He also can hold a good deal of sway in the political arena, because other elders are aware that the clan is truly behind the person.
Elders with sufficient power or influence in a domain may just declare themselves primogen. All that is required is the prince and other primogen acknowledge them in their position. The personal power required to be successful in this type of campaign can be derived from several sources - city elders owe you boons, you have a large sway over mortal institutions, or you are just basically liked and respected among all the city and court members.
Usually this method is only employed by very old city models, where the elders still believe that God will favor the righteous in a trial by ordeal or trial by combat. Newer cities where kindred are younger would scoff at this notion - it would be like awarding the presidency to the NBA's most valuable player.
"Its a dirty job but someone has to do it". In cities where there are engagements with enemies, no one wants to wear the shiney red bullseye, though many individuals will take up the cause because no one else wants it. Oftentimes, the person has been doing the job of primogen without the title - seeing to it that clan members get acknowledged, taking problems to the prince, etc. Eventually the title is tacked on, regardless of whether the person wanted it or not.
In many cities, a mix of all methods can be used for choosing primogen. Depends on the poltiical climate. In one city, a Prince may appoint a primogen of the malkavians, maybe because the malkavians can't seem to get it together enough to choose for themselves, the same prince may allow an electoral process to choose the Toreador primogen, and meanwhile the Brujah ends up on the council due to popular acclaim. Unless the Prince specifically states the rules of the game, all methods may be employed.
But theres a Catch.....
The Prince and reigning counsel can reject or accept the choice of Primogen. A Brujah that says he's primogen and gets all puffed up about it, may find he's excluded from counsel proceedings because the rest of the elders do not see him as a fit choice. This is probably a rare case, but can prevent a clearly disliked person from holding any true power in the city, even if the clan accepts him as primogen. A risky endeavor if the clan is powerful and large in number, because the counsel effectively cuts themselves off from that clan's resources.