τὸ κακὸν δοκεῖν ποτ᾽ ἐσθλὸν τῷδ᾽ ἔμμεν' ὅτῳ φρένας θεὸς ἄγει πρὸς ἄταν

evil seems good, soon or late, to him whose mind the god draws to mischief

Antigone, Sophocles

When the Gods set their sights on the destruction of a presumptuous man, the most potent tool in their arsenal is madness. This madness often manifests as a loss of reason or memory:

While the word was in the king's mouth, there fell a voice from
heaven, saying, O king Nebuchadnezzar, to thee it is spoken; The
kingdom is departed from thee...The same hour was the thing fulfilled
upon Nebuchadnezzar: and he was driven from men, and did eat grass as
oxen, and his body was wet with the dew of heaven, till his hairs were
grown like eagles' feathers, and his nails like birds' claws.

Daniel 4:31-33

Other times this madness can be a frenzied bloodlust, a boon to the noble hero at war that turns to curse at home in peace:

But now that he hath accomplished the labours of Eurystheus, Hera is
minded to brand him with the guilt of shedding kindred blood by
slaying his own children, and I am one with her. Come then, maid
unwed, child of murky Night, harden thy heart relentlessly, send forth
frenzy upon him, confound his mind even to the slaying of his
children, drive him, goad him wildly on his mad career, shake out the
sails of death, that when he has sent o'er Acheron's ferry that fair
group of children by his own murderous hand, he may learn to know how
fiercely against him the wrath of Hera burns and may also experience
mine

Herakles, Euripedes

Assume the Gods are bound by the laws of physics, our knowledge of science, and the biology of Man's brain. They must then, I suppose, have access to advanced technology (WARNING: TV Tropes Link!!!) to explain how they became gods in the first place.

Using their advanced knowledge of human biology, how can the Gods drive to madness those that they wish to destroy? The madness must be perceived by the victim's fellow man of antiquity to be of uncertain origin; i.e. no kidnapping and lobotomizing and returning the victim to society with a weird scar on his or her head.

Whom the Gods would destroy they first make mad.

The Masque of Pandora, Henry Wadsworth Longfellow

Meet the Salema porgy, also known as the dreamfish. It's a rather unassuming fish, found in the Atlantic Ocean and the Mediterranean. It can be seem near the water's surface, and to an untrained eye, in dim lighting, can look similar to its delicious (and oft-cooked) cousin, the gilt-headed bream.

This meek little porgy, however, can cause hallucinations that have been compared to LSD, with effects lasting for up to a day or two.

If you're a god, with sufficiently advanced technology - or, honestly, just someone who knows someone who knows a chef in the right restaurant - it shouldn't be any problem to substitute in a Salema porgy for a gilt-headed bream on someone's dinner plate. You can also choose from a number of hallucinogenic fish, if you want to go after someone from outside the Mediterranean.

Honestly, the same thing works with other organisms (Sasugasm mentioned one of a number of mushrooms that can do the same). Hallucinogenics aren't too uncommon, and very accessible for a god. They might look quite similar to any edible cousins they have. Simply slip the right substitute into someone's food, and you can produce hallucinations for days.

Your two first quotes have the answer.

WORDS

The god speaks to the king. The king does as the god told him so. Everybody thinks the king is mad; or, king really goes mad because the god tell him to do things that are awful.

The god can be Gaslighting anyone. Kill your son, kill all sons, kill your wife, order your wife to kill your son. It can be god himself as a voice in king head, burning bush, it can be god through a prophetess, it can be god in form of small animals that spell "kill", a lucid dream, a whisper in the dark.

You either go mad because god told you to kill your family because they want to kill you and after that you realize what have you done (Orestes in Aeschylus drama) or you go mad because you can't stop gods from telling you to kill everyone (As per Dr. Gregory House definition on when god talk back). And god can make a second feel like a thousand years. A thousands years that god can spend on telling you things you don't want to hear.

Mercury

The gods increase the amount of natural mercury in the soil near those they wish to destroy. The people inhale the vapors, they go mad.

No need for poison or advanced technology, all that is needed is to baffle them with bullshit.

People focus on what's immediately in front of them. Increase the trivia they need to sort through everyday. Eventually, they'll focus so much on the trivia in front of them that they'll forget there is such a thing as the big picture.

Next, control the data they're seeing. Carefully, move it gradually away from reality. Eventually, your victim will end up with a world view sufficiently divorced from reality that everyone else will think they're insane. This, in turn, will drive feelings of paranoia, as the victim believes they are proceeding in an entirely rational manner; it's everyone else that isn't seeing the world properly. Given enough time, the victim will simply disregard all opposing viewpoints as stupid or tainted with propaganda.

The rest is history. All you need is a handful of people with an agenda, no sufficiently advanced technology or poisons or anything extraordinary.

Sanity is based on a pretty narrow range of responses to sensory input. By making someone consistently receive different sensory input, or skew their responses to sensory input, you can make them insane.

Drugs (Psychedelics and hallucinogens)

There are a great many psychoactive drugs that someone could be subjected to. A continual low-grade dosing seems best (could be snuck into food, or could be injected as a device that has a slow-release mechanism to the bloodstream, or a genetically-modified parasite that produces psychedelic drugs as a metabolic byproduct).

Brain Modification (EM)

Transcranial Electrical Stimulation is a known technique that, effectively, stimulates or turns off bits of the brain via applied EM fields [sort of. This will do for our purposes].

Sufficiently advanced science gods could perhaps achieve precise effects (say, increased anger, lower inhibitions, lust...) using a remote antenna array aimed at an unfortunate soul's head.

Brain Modification (Chemical)

Emotional states are partly controlled via hormonal signals. Hormones could be thrown out of whack with methods that either supply a hormone directly, or that modify the human body to continually over/under-produce a particular hormone.

Science methods here can be genetically-engineered germs or parasites that produce the hormones (or a counter-agent that binds a particular hormone if you want less of it), or which alter a gland to change its hormone output.

Simple examples might be to have the hero permanently flooded with adrenaline like they are being chased by a bear, or to induce waves of lust at inopportune moments.

Change Reality (technology)

Something like small speakers that facilitate voices speaking to the hero out of nowhere could be used to continually push them in a certain direction, or make them doubt the reality around them. Move advance tricks could use holograms and full audiovisual simulations.

This is a case of a pseudo-god. Gods do not kill their subjects, gods work in mysterious ways, not boring shallow manipulations. Gods test, but do not kill. It is humans who kill, when they misinterpret their purpose in life. And I would think gods do not sit around worrying about how to control others. That is a trait of some type of inferiority complex, or personality disorder - not of a god.

Trying to find ways of diabolical killing is probably the nature of an insane person.

The gods could spray the mortal humans with hallucinogens, such as LSD, and then perform minor miracles such as projecting a hologram of a volcanic eruption to convince that mortal that he is a prophet of the coming volcanic apocalypse.