To become a founder of a family, you have to upgrade your townhouse to level 4. Afterwards, you can set up your lawyer. With a lawyer you can become a member of a family or start your own.

Open up the family menu by clicking on the family button on the left. To become a family member you have to apply for membership first. Send your application via the profile of your chosen family. You can find families by searching for them over the family search, or simply by clicking on rankings and finding them this way. If you wish, you can also apply for membership at several families at once. After that, you need to wait if or when your application will be accepted. 

Click on the pencil on the left side of the family screen to establish your own family. 

Use the global chat to recruit members for your family or ask players directly in a personal message.