How a primogen goes about getting business done, depends largely on the clan, and clans themselves have particular tendencies and views toward how primogen behave and how to manage their own clans.
Keep in mind that the primogen is not actually the defacto leader of a clan. The Prince is the ultimate authority in the city, and Primogen do not ~rule~ their clans persay, though they should and do have a good deal of influence over clan members (or as much as the clan gives them). A primogen can encourage their clans to follow an edict, even promote the prince's agenda with flowery speeches that convince the clan to follow her lead, OR the primogen may just deliver the message, and that's that. The primogen is not required to enforce traditions or punish clan members, that is the job of the Prince (or by extension the sheiff). This does not mean that primogen do not act as authorities of their own clans...some do rule their clans with an iron fist, and take it upon themselves to punish, berate or ostracize members, again it depends on the individual and how much authority he or she is allowed to have.
The primogen represents clan interest in the counsel, he advices clan members privately, he marshals the clan in battle, and, in the case of the mildest transgressions against the Traditions or the city's Kindred, a primogen can oversee the punishment of a clanmate. Any political power the primogen has comes from his clan. A primogen who is viewed as a leader and influencer of a large clan (and a large pool of resource) is much more likely to have pull over city politics than Joe Nosferatu who is a voice and not a leader for a group of sewer rats few people have ever seen.
Brujah tend to view primogen chairs as representative government. Brujah will often come to the table espousing some pet cause or project that he will not let another clan touch. Their politics seem to be more visionary than others. In dealing with his own clan the Brujah has an eclectic group of hot-tempered and often young charges who will not likely follow him if he's viewed as "the man". Though a fledgling who pushes her primogen's patience may find herself on the receiving end of an asskicking, unfortunately this usually doesn't teach the neonate much of anything- now they are more likely to rally other clan members against the primogen because of such "unfair treatment". To get around this by taking a whip much younger than the primogen who acts as a liaison or buffy between the primogen and the rabble. Often the whip and primogen feign some dislike for each other so that the clan members view the whip as a sympathetic peer, and not the primogen's lacky.
The madness of the malkavians does not make them inept or ineffective as primogen (in most cases). In fact, malkavians seem to approach issues with singular focus, and can be unpredictable in how they will vote or what cause they will pursue. Malkavians rely on omens and patterns that others do not see, and while a malkavian understands the implications of such visions, he may have a difficult time getting his peers to understand. As far as leading the clan, malkavian primogen have no standard or set way, and for the most part may leave their clan members to their own devices, yet stay in close contact, serving as representatives, rather than governors.
Some counsels will not allow primogen of the nosferatu to take their seats at all, and while the reasons sited are flimsy reasons (personal hygeine included). When they do come to the table, they are most likely to have the best information, and they are just as likely to not share it or only share portions of it. Nosferatu often work for their own agendas, the pursuit of more information and leave the rest of the politicking to the other primogen. In dealings with their own clan, Nosferatu show the least concern for titltles and formality. The primogen is seen more as a knowledge resource than a political authority or leader. The nosferatu has to deal with problems other clans do not, as clan members are often limited in their access to elysium or other kindred functions. Because many nosferatu are hermitlike, representing one, can be problematic, as other primogen may not even know the man in question, at this point, the primogen can rely on boons gained from information trading to save his clan member. Since boons and information are the currency of the nosferatu, the primogen often serves as the director of his clans information gathering, so as to garner influence through these boons.
Toreador primogen are some of the busiest socialites in elysium, they keep in touch with everyone and try to learn everything about the views of others before stepping into the counsel chamber - in truth often having their finger on the pulse of the kindred population, they know who's screwing who, and who's screwing who over, and what causes are likely to move the population as a whole. Toreador primogen often have a great grasp of the hidden social interactions that are behind all the exterior moves, and they are adept at stirring and settling the waters of the primogen chamber. The Toreador primogen leads by subtlety, protocol, and etiquette, and the unofficial hierarchy of the clan is determined largely by social status. These social nuances can be applied to shephard the clan (or all kindred) in whatever direction the primogen deems necessary.
Tremere often view the primogen position as a stepping stone to higher power (princedom?) though it may also be a respectable position to hold once the glass ceiling within their own clan has been reached. They are often the ones that know the most about the regent (though regents sometimes serve as primogen) and can use this close association to their advantage. They are often the least tolerant of idiocy and will come to the counsel chamber prepared and with notes in hand. The leading of the tremere clan is a relatively simple thing, they often have a clear and concise set of guidelines to follow, which are in order with the tremere clan's methodology. The role of the primogen is clear: the primogen speaks and the clan members obey. The dynamic between the regent and the primogen can become a tricky thing. The primogen outranks the regent in social politics, but the regent outranks the primogen within the clan, setting the state for intense, if subtle, conflict.
Like Tremere, Ventrue also tend to view the primogen chair as a stepping stone to greater things. The chair is also a place for a Ventrue to return to after leaving princedom (provided she wasn't removed in a coup). The Ventrue are seen as pillars of the camarilla, and as such they cannot be seen engaged in unsordid shenanigans, and often Primogen of the Ventrue are very old and conservative. When dealing with their own clans, Ventrue take a governor, rather than representative, position. The ventrue primogen is typically one part field marshal, two parts corporate manager, and one part charming master of the fine art of schmoozing. There is no tactic that the Ventrue primogen will not adopt to ger her way, support her clan and further her standing in the camarilla. Within the clan, the ends often justify the means, and focus is placed on success (and not the manner with which one achieved the success). Ventrue weild influence very effectivly in the game of politics, and the best way for other clans to make sure they don't lose, is to not go against the Ventrue.
Gangrel primogen are seen in cities that host a large gangrel population, usually areas that are surrounded by enough space and wilderness to host such a large population. A gangrel on the counsel may seem like an oxymoron, but it would be folly to assume that their wandering ways makes them stupid. Many gangrel have a rich education due to their traveling. Most gangrel primogen tend to be younger and visionary. Older gangrel usually have left the trappings of politics behind, and their appearance may be disturbing to others. As primogen, Gangrel are usually on their best behavior, feeling the need to convince others they are still a faithful part of the sect. Here and there throughout the Camarilla are those gangrel who remain affiliated with the sect solely for the boons they receive in exchange for their martial abilities. Some even say the warlord Karsh is the ultimate example of this approach (though they don't say it too loudly). Gangrel primgoen are as laissez-faire as they come and are about as antiheirarchical as they can get away with. A Gangrel primogen is as likely to hold his seat for telling enthralling tales as for his combat prowess. One rare occasions when a Gangrel primogen needs to act as an authority figure for his clan, its likely to take the form of a blunt, no-nonsense statement of fact. "Stop hunting in the Malkavians' feeding grounds or the prince is going to stake your ass. Don't say you weren't warned." One the message is delivered, the primogen shrugs off the mantle of authority and reverts to the role of solitary predator.