Mazal tov on the birth of your baby!
Optimally, the bedikah of the hefsek taharah and one bedikah during the seven clean days (ideally on the first day) is as deep as you can go without hurting yourself. The other bedikot should be thorough, but you need not make as much of an effort to go deeply. When performing bedikot, you should insert your finger very gently and slowly, rotating into all the crevices. There is no fixed measure of exactly how much of your finger must be inserted.
The postpartum recovery period, however, is a special case. You should take extra care to be gentle, still making your hefsek taharah at least to the depth of your first knuckle. Performing it while lying down and slightly moistening the bedikah cloth with water may prove helpful. You may also omit the moch dachuk while postpartum sensitivity continues.