Instructions
Walk up to Dogmeat. When Dogmeat becomes your follower, he typically stays within a few feet from your character at all times unless you manually tell him to stop or he becomes engaged in battle with an enemy or monster.
Press "X" to talk to Dogmeat. Select the "Let's Exchange Equipment" option from the menu.
Give Dogmeat any food that you have. This can be drink products such as beer or Nuka-Cola or food such as meat from animals and irradiated monsters you've killed out in the wasteland. Highlight the item from your list and press "X" to give the food to Dogmeat.
Press "Circle" to exit the menu. Dogmeat will take the food items that you've given him and begin to eat them.