Depending on how fearsome of a reputation he wants and if the enemies of his people are downwind of the island, he can cause the island to disappear beneath the waves with 1960 era technology.
Not those piddly little nuclear devices mentioned in other answers, but this:
Castle Bravo was a thermonuclear device which, due to a misunderstanding of how lithium reacted under the staggering conditions in the secondary, achieved a yield of 15Mt, the largest nuclear device ever created by the United States. The device destroyed almost all the surface vegetation and structures of the surrounding atoll, and excavated a crater 1.8km in diameter.
Oh, wait. The island is 11km by 4km. Maybe the Russians can help:
Tsar Bomba "on the waaaaaay"
This little beauty achieved a yield of 50Mt, without its third stage (the tamper was made of lead rather than Uranium). The fireball alone was five miles in diameter, so an airburst will essentially vapourize anything on the surface of the island. I haven't found any reliable estimate as to the size crater a ground burst would create, but it would likely cover a fair fraction of the island. IF you were particularly bloody minded, use the Uranium tamper and bring the yield up to 100Mt.
"But that's hardly elegant", you protest. Well, the ever inventive Americans may have discovered ways to reach even higher yields (gigaton) without the gargantuan "steampunk" like design of the Tsar Bomba:
But given that high efficiency is tied to high yields — and relatively high weights — it’s clear that the innovations that allowed for the placing of warheads on MIRVed, submarine-launched platforms are still pretty impressive. The really magical range seems to be for weapons that in the hundred kiloton range (more than 100 kilotons but under a megaton), yet under 1,000 kilograms. Every one of those dates from after 1962, and probably involves the real breakthroughs in warhead design that were first used with the Operation Dominic test series (1962). This is the kind of strategic miniaturization that makes war planners happy.
Or, to take another tack, and returning to the initial impetus for me looking at this topic, we know that the famous “Tsar Bomba” of the Soviet Union weighed 27,000 kilograms and had a maximum yield of 100 Mt, giving it a yield-to-weight ratio of “only” 3.43 kilotons/kilograms. That’s pretty high, but not for a weapon that used so much fusion energy. It was clear to the Atomic Energy Commission that the Soviets had just scaled up a traditional H-bomb design and had not developed any new tricks. By contrast, the US was confident in 1961 that they could make a 100 Mt weapon that weighed around 13,600 kg (30,000 lb) — an impressive 7.35 kiloton/kilogram ratio, something well above the 6 kt/kg achieved maximum. By 1962, after the Dominic series, they thought they might be able to pull off 50 Mt in only a 4,500 kg (10,000 lb) package — a kind of ridiculous 11 kt/kg ratio. (In this estimate, they noted that the weapon might have an impractically large diameter as a result, perhaps because the secondary was spherical as opposed to cylindrical.) So we can see, without really knowing much about the US had in mind, that it was planning something very, very different from what the Soviets set off.
Still not good enough?
To eradicate any trace of the island (or indeed any nearby continents), the most powerful single "plausible" device was outlined by Anthony Zuppero, who was apparently asked to do the calculation for a device which would wipe out the Soviet rocket fields in Siberia in a two minute window. He evidently calculated a massive 5Gt weapon launched by an ORION nuclear pulse drive...
Doomsday ORION weapon, artwork by William Black