To buy a brand new faction, go to: City > Create A Faction which i located at the bottom left part of the city, under Factions. A new faction starts in 'peaceful' status, during which it can't be attacked. This status changes if you declare war on any faction.
Specs of a newly bought faction:
Holds: 5 Members
Costs: $1,000 x current number of factions in the game
Factions lose a respect point every 24 hours.
If your respect hits zero points, you will lose your faction.
You can gain respect points by doing "Organized Crime" with your faction, but they require at least 2 members (certain crimes require more).
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