C7 or ‘C Dominant Seventh’ can be a really handy chord. It is formed by adding your fourth finger to the C Major chord. When attempting to get to it quickly it can be helpful to form the C chord first and then add the fourth finger.
It can be really useful when thought of as a moveable shape. Move it up two frets and it becomes a D7 chord, up another two frets an E7. As a moveable shape it becomes a four string chord so strum from the fifth string and lean your first finger over to deaden the first string. Play just the middle four strings.
Try it playing Blues in G using G, C7 and D7 - with D7 played as the C7 shape just moved two frets higher. Notice how I can get a nice staccato effect as all the strings are fingered - no ringing notes. Finally, you can add some really cool alternate bass notes by swinging your third finger over to the sixth string.
There it is! C7. A handy chord with lots of uses.