No Class/Hero XP and no Skill points

You will not have the option to convert the crafting parts in to skill points until after the next update in April.  For now, the only way to upgrade your cards is by playing the particular class and earning XP.  Once you earn enough XP, you will level up and be awarded a skill point.  You can then use the skill point earned in that class to unlock/upgrade a star card in that same class, assuming you have met the appropriate level gate.