The easiest way is with a bit of simple modding.
First, open the raws folder (DF main directory > raw > objects), and open the creature_standard text file.
The dwarf entry is first, but you'll have to scroll down a bit to find the dwarven attacks. Here they are:
[ATTACK:PUNCH:BODYPART:BY_TYPE:GRASP]
[ATTACK_SKILL:GRASP_STRIKE]
[ATTACK_VERB:punch:punches]
This means that the attack will use as much of the available tissue as possible, rather than, say, thrusting with a spike.
[ATTACK_CONTACT_PERC:100]
[ATTACK_FLAG_WITH]
--- [ATTACK_PRIORITY:MAIN]
[ATTACK:KICK:BODYPART:BY_TYPE:STANCE]
[ATTACK_SKILL:STANCE_STRIKE]
[ATTACK_VERB:kick:kicks]
[ATTACK_CONTACT_PERC:100]
[ATTACK_FLAG_WITH]
--- [ATTACK_PRIORITY:SECOND]
This causes all of the nails on the finger's of a given grasp to be used.
[ATTACK:SCRATCH:CHILD_TISSUE_LAYER_GROUP:BY_TYPE:GRASP:BY_CATEGORY:FINGER:NAIL]
[ATTACK_SKILL:GRASP_STRIKE]
[ATTACK_VERB:scratch:scratches]
[ATTACK_CONTACT_PERC:100]
[ATTACK_PENETRATION_PERC:100]
[ATTACK_FLAG_EDGE]
--- [ATTACK_PRIORITY:SECOND]
This causes all of the teeth on a given head to be used.
[ATTACK:BITE:CHILD_BODYPART_GROUP:BY_CATEGORY:HEAD:BY_CATEGORY:TOOTH]
[ATTACK_SKILL:BITE]
[ATTACK_VERB:bite:bites]
[ATTACK_CONTACT_PERC:100]
[ATTACK_PENETRATION_PERC:100]
[ATTACK_FLAG_EDGE]
--- [ATTACK_PRIORITY:SECOND]
[ATTACK_FLAG_CANLATCH]
I put --- in front of the relevant entries. As far as I know, creatures will always use their MAIN attacks if they can, and only use their SECOND attacks if no main attacks are possible. Just change [ATTACK_PRIORITY:SECOND] entries to [ATTACK_PRIORITY:MAIN]. Caps are important.
Now dwarves in newly genned worlds will have a chance of using each MAIN attack. Raws are now saved with worlds, so if you want adventurers to bite in an already genned world, you'll have to change their raws in the same way. (DF main folder > data > save > regionN > raw > objects)