In order to summon the Harvest King you need to raise your Farm Degree to the level of "hero" or above. Each day the profit you earn from Crops, Animal Products, Fishing, or Mining are added together in each category. At the end of the season you will earn a Farm Degree title depending on which category you earned the most money in. You can find your current Farm Degree rank by going into your Farm Summary menu and looking at your Player details.
To earn a hero rank you need to earn a lot of profit in one of the four categories (crops, ranching, fishing, mining). It doesn't matter which category you choose to become a hero in, but only the items you ship in the current season (28 days) count towards your Farm Degree
getting the hero rank in shipping crop is probably the easiest way to do it. You don't even need to buy extra crop lands from Hamilton. To get the hero rank in crops, you'll need to earn at least 100 000 G in the season you're in.
The easiest time to do it is during summer of fall. During the beginning of the game, you get these cherry trees that can be harvested during the summer. Cherries cost quite a lot. You can also plant crops such as corn, tomatoes, sugar canes, watermelons, and more. Tea leaves are a recommended choice since they can be harvested in 3 days and cost a lot.
Fall is a REALLY good season to earn your hero rank since you can plant these really expensive flowers called the "blue mist flower". These flowers will sell for the same amount of profit no matter what quality of soil they're in (so no decents! in a way, they're ALL shining!) The only thing soil quality affects is the speed of which the flowers mature. The flowers cost 50 G a bag and is sold at the Marimba Farm. So, just plant a lot of blue mist flowers during the fall, and you'll earn your hero rank. (108 bluemist flowers costs around 379, 080 G!)
To earn the hero rank, you'll need to earn at least 60, 000 G in profit of all your shipped animal products. For this, i suggest that you ugrade your barn and coop to the max. level upgraded (lvl 3) if you can since this will be the easiest way for you to do it. I guess the most profitable livestock are cows and goats since they can produce milk everyday, which you can turn into cheese. Cheese cost more than butter. But doing this is pretty risky since you need a certain quality in order to reach the 60 000G. (so, NO decents, just good, perfect, or shining)
You need to earn at least 300 000 G to earn the hero rank. But this is pretty hard to do and is not recommended unless you are really pro. at mining and have good luck since you actually need to process the jewels and ores to earn the money. Processing the wonderfuls and ores aren't like in Tree of Tranquility where you just find wonderfuls and ores with the sparkly things, but you need to be lucky to get the expensive jewels. Remember, the things you refine will be like 80% junk (glass, junk metal...) and 20% jewels.
Fishing requires less than mining, but you'll need to earn at least 40 000 G to become a hero. This might seem to be a little, but remember, most fish that you catch at certain places can sell for at little as 7 G each. The only fish that sell for anything good is just like maybe the King Fishes, which, are extremely hard to catch. But if you really want to earn your hero rank in fishing, i suggest you go to Toucan Island and fish along the beach. You need to have a high level fishing rod to do this though. (iron or copper won't get you anything good => need a silver, gold, or goddess level to do this) But another good thing is that if you have a husband/wife and children, you can get them to fish with you. (Like every morning, you can choose a chore for them to do. If you choose 'fish', then they'll fish for like 6:00 - 18:00. The money they earn from doing this will be counted at the end of the day)
AFTER EARNING THE HERO RANK
When you go to bed on the last day of the season (the 28th day), your farm degree will be calculated and there's FINN (oh goodie! ) who tell you what rank you got. You also get a prize for your hard work. Like for example, in crops, if you earn the hero rank, you get 70 good fertilizer! If you made it then, congratulations~, but if you didn't, then try again the next season and work even harder. There's no time limit, so the Harvest Goddess won't actually die.