How to cut a cooked chicken, including the bones, with a cleaver?
I think what you want to do is this, cribbed from this book, which has an excellent step by step guide with pictures:


Remove the wings. Stand the chicken on its neck, legs point to the ceiling and breasts pointing away from you. slide the cleaver between the body and wing, cutting through the joint close to the body.  You should slide the knife between the joint cutting only the tendons, not the bone.
Remove the legs.  lie the chicken on its back one leg near you, one leg away from you.  slide the knife between the joint near the body, again cutting only tendon not bone and remove the leg.
Separate the thigh from the drumstick, again go for the joint.  If you want the legs to be in smaller pieces then you can cut each thigh and drumstick in two with a firm action.  You have 2 choices for technique


Start with the cleaver tip furthest from you on the board, place the thigh under the blade and push down firmly and quickly.  you should cut through the bone.
Chop through the bone by lifting the cleaver in the air and bringing down hard and fast.  This has better chance of a cleaner cut through the bone, but you need to be a better aim. 

Split the carcass in half.  Stand in the same position as 1. and with the blade parallel to to your body cut down through the carcass so you have the back and the breast in separate pieces
Remove the breast from the bone by sliding the knife (might want to use a smaller knife here) between the meat and the bone.  
Remove the backbone out of the back, kitchen scissors are easiest.  you can skip this if the back doesn't have much meat.
Cut the back pieces into nice size pieces across the length.
Slice the breast pieces crosswise into nice size pieces
Reassemble the chicken on the plate with the breast meat on top of the back pieces


this picture doesn't really do it justice, but until I can find a better one...


this video shows it being done with a bit more force and not for presentation, but might be more what you are after