Zordnil's solution is great for explaining how the heart system works in Stardew Valley. However, it does not really explain the fastest way to get all of the heart related achievements. The easiest and best method, in my opinion, to get this achievement is to get the recipe for Sashimi.
First, you must have an upgraded house with a kitchen. This will cost you 10,000 gold and 450 wood. The best way to get this is to fish nonstop and cut down trees in the forest south of your house. While doing this, you can start working on getting the actual recipe unlocked.
The Sashimi recipe can be obtained by reached a three heart level friendship with Linus, the man who lives in the tent to the north east, right next to the mines. You can give him many things, but the easiest would be yams in the fall because you can grow them. Note that you can only give him two a week, so either continue playing normally or sleep until the next week is available. If it is the third of winter, you can give it to him for a huge bonus as this is his birthday.
After you have a three heart friendship with him, he should send you the recipe the next day through the mail. All this recipe requires is one fish. So the whole reason behind making money with fishing is that your fishing level will be higher, which will let you catch fish easier. Also, you can purchase bait from Willy at the beach for 5 gold a piece if you have the Fiberglass rod (which will cost you 1800 gold at Willy's shop).
Once the recipe is obtained, all you have to do is fish and fish all day, then go into your kitchen and turn it into Sashimi. This meal is loved by Sebastian and liked by everyone but Willy, Emily, Evelyn, Kent, and Krobus.
So, essentially all you have to do is make sure that every week you give 2 Sashimi to the villagers you have decided to use to get this achievement. Remember to give them it on their birthday as well and you will have this achievement as well as all the other heart related achievements in no time.