I know that according to the whitewolf wikia "They Embrace exclusively within their family and are heavily focused on the goals of money and necromantic power."

But you maybe can start the game as a "outsider" and only be part of family after have solved some important things and even inside the family suffer a some discrimination, but i really wanna play as a giovanni clan member.

EDIT > a mod that allow you to play as a follower of set is a welcome suggestion too.

EDIT 2 > Aparently the unique mod who allows you to play in giovanni clan is Antitribu BUT Antitribu makes some quests very annoyng...

You are looking for the ANTITRIBU mod. It is currently in 1.1 version, but don't forget the hotfix, as well. Can be applied over gog's version just fine, without issues at all. You play as:

1. Giovanni
2. Samedi (instead of Ravnos)
3. Nagaraja (instead of Assamite)
4. Follower of Set
5. Lasombra
6. Tzimisce (Auspex, Animalism, Vicissitude)
7. Old Clan Tzimisce (Auspex, Animalism, Dominate + Koldunic Sorcery)

It is still under construction, with a promised Sabbat questline and ending even, too. A very fun mod to play, although it has its fair share of issues and bugs. Some players report that Koldunism doesn't work in 1.1, keep in mind. A great mod overall. You HAVE to play it!

I appreciate the hard work to make this mod BUT there are a lot of quests that you get no clue/info about what you need to do like the "blood boss" in Museum or the ship quest, so this mod is unplayable for me...