Even if your family fill all the senate offices you won't have a majority. There are 200 to 300 senators depending upon the date and the senate officers aren't their leaders, they're meant to be the agents of the senate's will.

In game terms, you can't really do much to influence the senate in your favor, in the end they're out to get you so your best bet is just to increase your popularity with the people by conquering new territories and then once you have their favor go stomp the senate into the dirt.