It depends on your play style. Since characters generally have their own kind of play style themselves, you don't have to go leveling up different characters to get similar skills.
For you and for future reference as well:
Barry has all the Weapon Master skills, so if you like ranged weapons level up his skills.
Jill has dodging skills, so level up her skills if you get hit too often.
Moira is awesome with melee, so if you love to kick monsters in their faces and do a lot of damage level her up.
If you aren't good at dodging, Gina has a lot of health skills, so if you need to survive better level up her skills.
If you love throwing stuff at enemies, Evgeny has all the bottle/grenade skills, both offensive and defensive.
Chris has skills that defend him and block damage, so his skills are another alternative to Jill's dodging and Gina's healing.