To unlock the Forza Driver's Cup you must first complete the Masters Championship.
To win the Driver's Cup you need 50,000 series points. Each series is worth 10,000 points and each showcase is worth 2,000. There are 11 series and 4 showcases to choose from, however, unlike the previous championships there are not enough showcases for them to make a meaningful contribution to your series points (provided you gain maximum points from each series). So the quickest way to complete this championship is through winning 5 series.
Once you have won the Forza Driver's Cup, you will unlock the additional 'Elite' events in each championship required to get the 'Fully Completed' achievements for each Cup.