The basic principles of Hellenic Polytheism

Xenia

The rule of hospitality. Treat your guests with kindness. Be kind and generous to your community and strangers. Xenia is protected by Lord Zeus Xenios (Latin: Iuppiter Hospitalis) Xenia is held sacred, especially if the stranger comes to your door, because sometimes Gods may disguise themselves as mortals. Since the person standing before you may be a God, we should treat everyone with great kindness and generosity. Such compassion for the vulnerable is the desire of King Zeus. The stranger (Xenos) is bound to his host, and both the Xenos and the host are bound by obligations.

Katharsis and Miasma

Miasma is the impurity we gain throughout our lives. It is both the spiritual impurity and the actions within the material realm that contaminate us. However, miasma is not "sin", and it doesn't mean that our world is impure. Miasma is the result of actions against Themis (Natural law). Childbirth, menstruation, and natural death are all processes which disrupt Natural Law. Other things that are miasmic are things which result in the disruption of Themis, like dirt, sexual fluids, sickness,... All of these are associated with minor miasma. Things that cause major miasma are more serious. They cause a stain that takes more than purification to get rid of. Major miasma can be caused by things like murder, incest,... Miasma can be dangerous, since it can cause prayers and rituals to become null, and instead it places us in connection to the spirits of punishment.

“Do not pour libations of sparkling wine to Zeus and other immortals at dawn with unwashed hands. They do not hear your prayers, but spit them back.” -Hesiod

Katharsis is the state in which you approach the gods. It is the state of ritual purity. Katharis is essential to maintain during worship. It is not just confined to yourself, but it is also relevant to the harmony and stability of society.

So, what can you do to cleanse yourself to achieve ritual purity? First of all, very basic hygiene. Take a bath, shower or wash your hands before engaging with a sacred space. However, to fully achieve Katharsis, one would need a Khernips. A Khernips is essentially "holy water" used to ritually cleanse one's hands and face. It is made with the most fundamental elements: Fire and Water. This is how you make one:

Get a bowl of water. Ideally flowing water (sea or river water)
Extinguish fire in the bowl. You can use a match, incense, dried herbs or even a twig. When dropping it in the water, you can say "Xerniptosai" (Be purified)
Wash your hands and face with the water. You can say "Kherniptomai", which means "I wash with lusteral water".

You can also use this water to cleanse your altar or objects you place on it.

Arete

Your personal best. The Gods expect your personal best, not someone else's. Try your best and they will be satisfied.

Eusebeia

The principle of Piety. It is the act of showing respect and reverence to the Gods.

The Terminology

Theoi - Our Gods

Orthopraxic - this religion is based on your deeds, not dogma.

Kharis - the bond and connection you build and strengthen with the Gods. You strengthen this bond by worshipping and praying to the Gods.

Altar - Where you pray and leave offerings. An altar is not mandatory, and can be as personal as you want.

Lyma - The day-to-day dirt you accumulate through the day.