What is the process of adding a amendment to the US Constitution?

The amendment process has two steps, proposal and ratification. The amendment must first be proposed a 2/3 vote of congress or by 2/3 of the state legislatures.

Ratification occurs with a 3/4 vote of the state legislatures or a 3/4 vote of state conventions made up of delegates (not necessarily legislators) elected by the people. See the related link for more detail.