Why does the turn coordinator ball move to the right during a climb?
There are 4 left turning tendencies in aircraft with "clockwise" (as seen from the pilots seat)
turning propeller, 3 of which may apply during a full power climb out.  The four are:


Prop torque  - prop turns right, plane rolls left - more noticeable at high power/slower 
  speed
Gyroscopic precession - nose goes down - plane yaws left - not a factor in a climb - some 
  effect as plane levels out, also there on tail draggers as rear of plane rises during take off 
  roll
Asymmetric prop blade thrust at higher AOA (p factor).  Differences in thrust of ascending
 and descending prop blades will cause yaw to the left.
Spiral slipstream striking vertical stabilizer and side of fuselage, causing yaw to left.


Number 1 has to be watched in higher performance engines during takeoff roll.  Lifting off with insufficient airspeed will cause the plane to roll over and crash.  Fortunately, in a Cessna 172, this tendency is much more modest but should not be ignored.

3 and 4 are probably the main culprits of left yawing tendency in the climb.  The answer:  pitch to safe climbing airspeed (generally Vy) at full power and "step on that ball" with 
right rudder.

Post script:  the rolling tendency caused by high power lower speed torque may cause the pilot to feed in more right aileron when the better input is coordinated with rudder.  As the plane accelerates from rotation and liftoff speed to Vy, it would be easy to miss you've rolled slightly to the right due to increased aerodynamic force.  In straight line flight this would not be a major problem (unless you were dangerously near aileron reversal stall speed).  I would work on glancing down at the ball as soon as safe climb is established, perhaps while checking airspeed.  Stay coordinated.