"Do things right" means to do something the proper way, following instructions, as per the rules.
"Make things right" implies that you've upset someone or done something wrong, and you now want to make reparations. For example, you borrowed money from a friend a long time ago, and completely forgot about it. Suddenly you find the receipt from that money among your old papers, and you're embarrassed that you forgot your debt. You apologize to your friend, and tell him "I want to make things right. Here's the money I owe you, with a little extra so that you can take your wife out to dinner at a nice restaurant as my guests."