Is weekly heroic strike progress across account or per character?
This would mean that I can play the strikes one time per character and get the reward 3 times in a week
You can play the weekly strike once per character and receive rewards on each character - so if you have 3 characters, you can get up to 27 coins per week, if you can do it on the hardest difficulty with each.
You can play the weekly heroic strike multiple times per character but you can only get the rewards once per difficulty level per character in my experience.
There are three difficulty levels (Level 22, Level 26 and Level 28) each of which has slightly different mission modifiers. The harder levels obviously yield larger rewards. So in principal you can get three sets of rewards per character if you complete all three difficulty levels.
When you are selecting your destination and are choosing the difficulty level hovering over the activity name will tell you what rewards (if any) you are eligible to receive from that activity.
Btw I am not sure whether doing a harder difficulty level first prevents you from getting the rewards for the lower difficulty level.