Question: I know you get 2 set pieces in the new season for reaching level 70, what else do you have to do to get the other 4 pieces?

Can you get a free set again if you complete the parameters with a new character?

Answer: 
You get two more set pieces for completing the Chapter 4 objective "Mercy" (Defeat Zoltan Kulle on Torment II) and two pieces for completing the Chapter 4 objective "Great Expectations" (Complete a Level 20 Greater Rift solo).
As yuuki didn't answer this: 

**NO**, you can't get the items on other characters.