How does one become a member of the DNC?

Three ways: by election, by appointment, or by virtue of an office they hold.

Election: Each state is allotted a certain number of members to the DNC. These slots are filled by election at the State Assembly and Convention in Presidential election years.

Appointment: The Chair of the DNC has the authority to appoint at-Large members, usually selected to address a particular topic.

Virtue of another office: The Chair and Vice Chair of each state Democratic Party are automatically members of the DNC. This is also true of the leader of several party-based organizations such as the Democratic Legislative Campaign Committee, the Democratic Governors Association, the Democratic Senatorial Campaign Committee, etc.