There are two requirements you need to fulfill in order to 'fully complete' a Championship - you need to complete all the events in the Championship, including the Elite events that only unlock once you win the Forza Driver’s Cup, and you must obtain a minimum number of Series Points from these events. This means you do not have to win every race in the Championship, just meet a certain SP threshold.
For the Evolution Championship, you need to complete 13 series and 5 showcase events to 'fully complete' the Championship, scoring a minimum of 22,000 Series Points along the way.