“Erik the Slayer” is a Local Quest in The Elder Scrolls V: Skyrim
Wandering around Rorikstead you will find Erik and his father Mralki. Talking with Erik, he will notice you are an adventurer and ask you to help convince his father to let him go out and adventure. Agree and you will be sent over to the Frostfruit Inn.
Inside the Inn you will need to talk with Mralki. Save and then talk with him. After the introduction you can either bribe (330 gold), persuade or intimidate Mralki to take Erik to Whiterun to be fitted for armor. With a good Speech Skill (17+), all options should be available with a good chance of passing.
When you succeed in any of the 3 ways to get the armor for Erik just head back to him. Talk with him and tell him the good news.