Put your thumb on C, your pointer finger on Eb, your middle finger on F, and your ring finger on F#.
Play all four notes going up, one at a time, and then back down: C, Eb, F, F#, F, Eb and C. Practice going up and down until you are comfortable with the bottom half of the scale.
Let your thumb swing under your hand as you go up the first four notes of the blues scale: C, Eb, F and F#. As you play F#, let your thumb hover over the G.
Play the top half of the scale: play G with your thumb, Bb with your pointer finger, and C with your middle finger.
Practice going up and down the top half of the scale: G, Bb, C, Bb and G, until it feels easy.
Combine the bottom half and top half of the scale, using the fingering you learned above, to play the entire C Blues Scale: C, Eb, F, F#, G, Bb, C, Bb, G, F#, F, Eb, C. Remember to swing your arm out so your thumb can get under the hand to play the G. On the way down, angle your arm in the same position to cross the fourth finger over the thumb when you play the F#. Keep practicing until the C Blues Scale feels comfortable.
Mix up the notes in the C Blues Scale to make your own blues melody. You can play them in order or out of order. Experiment with playing the C Blues Scale notes all over the piano: up high, in the middle and down low.