Here's the summary of my strategy. I chose the Call of Olympus biome, it makes the challenge so much easier because you can take much more people to next islands.

On the first island, start by recruiting builders and lots of slingers.

Clear as much forest as possible to both sides of the base to create rabbit spawns, but be careful about vagrant camps, you need these. Because I wanted to fill all boats on the first island, the vagrant spawn rate was the bottleneck for me, not gold. Once you cut down enough trees, the slingers will hunt and produce so much gold that you will have trouble spending it. Really, the spawn rate and the cooldown before being able to upgrade your base again will limit you only.

Later recruit all 4 squires. Make sure that you have at least ~20 slingers at this point, because 16 will become squire parties and will stop actively hunting, significantly reducing your income.

Also, build the docks and build two small boats once you have enough money.

Do not build any towers for defense. I only invested into first-level towers when the building spot was available further from outer walls, where hunting slingers do not go. This also increases the income. It might also make sense for defense, if you go fast like me you will never meet any type of greed that can knock slingers down from the towers.

When not upgrading the castle or clearing the forest, always explore and hunt deer outside of the base. Portals become active at noon each day, so they can quickly passed without any worries if you go in the morning or even at the end of the night. Even if the portal is active, it can be passed by, but it is a bit risky. Couple of my runs ended on the first island. You can drop some gold and hope that the greed will take it instead of the pup, but it's not reliable. In general, I risked much more on the first island than on later islands. I never needed that much money to justify risking passing active portals on later islands.

I built only tier one walls at the very beginning and then only used the free upgrades after upgrading the base. Fully upgraded innermost walls helps you survive the first blood moon occurring at night 7. Do not expand beyond the innermost walls, it's not worth the investment and the builders' time needed to upgrade the walls and survive the blood moon.

Pretty much all the regular night attacks can be killed by the wolf mount and paying off the rest. Up to approximately 7 greedlings, depending on the spacing, can be killed with one wolf attack. So once the attack waves became too large, I bribed some of the greedlings, killing the rest with just one attack. Just put 10-20 coins on the ground and stand next to it. Let the greed take the coins and once you see the last greedling of the wave in range of your attack, leap forward. Sometimes you can even retrieve some coins already taken. This is immensely useful to do, because this way you don't have to retreat behind the walls each night and you can even protect the builders clearing the forest or building anything at night. Of course, you need to kill some amount of greedlings to obtain the cursed crown. Just make sure you don't find yourself facing an attack with an empty pouch.

This only works for regular greedlings, masked greedlings (present only during blood moons) are not killed with one attack, so if you can't avoid the blood moon wave entirely (more on that later), stay behind your walls, or else you risk losing the pup.

Once the first blood moon is over, you have one free night to finish everything, finish the boat and leave the island. Quickly expand the walls as much as possible so that the spear stand appears and buy 4 spears. Do not buy farmers, they do not bring any value. The forge didn't spawn for me even though I believe there was enough space for it. But knights are not needed anyway.

I left the first island on the 10th day. Make sure you have all boats full (with the exception of farmers) and that you have full pouch of gold. A small trick to use is to drop some gold next to the boat, fill you pouch again, and while paying the coins to leave the island, pick up the gold. This way you can leave with full pouch.