I was wondering what the beliefs of the Byzantine rite are about the afterlife... What exactly are the natures of Heaven and Hell?
First, you might want to learn the proper way of phrasing things. It's not "Byzantine Rite." It's the Eastern Church (for both Eastern Catholics and Eastern Orthodox), or it's each group by name.
As for the "exact natures" of heaven and hell, these are mysteries that the Eastern Church doesn't try to probe too deeply with our minds. That is because we recognize that the human mind has limits, and to try to push those limits can lead to a loss of spiritual focus. In other words, the Eastern Church has an apophatic mindset. It is often more accurate and true to say what something is not, rather than say what it is (when the something in question is beyond ordinary space and time).
So it is with heaven and hell. Clearly they are not places on earth, in or above the earth, or other planets. Those kinds of ideas may have been popular when science was much more primitive, but they are clearly inaccurate and obsolete now. However, heaven and hell are not merely metaphors for our states of mind; nor are they even metaphors for our moral choices and relationships.
Perhaps the most that can be said about heaven and hell are what Jesus Christ said about them.
Heaven is the Kingdom of Heaven, the Kingdom of God, the mansion with many rooms of the Heavenly Father. It is eternal life, and it is the presence of Christ, for Christ declared that He is the resurrection and the life. It is also like a city, but fantastically superior, wherein the righteous (human and angelic) dwell with God and where God dwells with us, and where every tear shall be wiped away. Heaven is not saying "Lord, Lord" or working miracles in Christ's name; it is, instead, doing the will of the Heavenly Father. It is the paradise that was prepared for us since the foundation of the world, for "When I was hungry, you gave me something to eat . . ." Heaven is accepting the presence of the Holy Spirit within us and among us; and thus heaven is not blaspheming against the Holy Spirit. Heaven is thus in our heart and in our relationships and in the neighbor, for Christ taught that the Kingdom of Heaven is within you and among you and "whatsoever you did to the least of My brothers, that You did unto me." Heaven is also the Eucharist, for Christ said "This is My Body . . . this is My Blood." Finally, as St. Paul observed, heaven is beyond the capacity of human senses and human words, but (as he and St. John observed) heaven is seeing God directly. And God is love, as St. John also observed. In short, heaven is living in communion with God --directly and with the neighbor, starting now and foretasting forever. For God is good and He loves mankind.
Hell is rejecting all that: starting now and foretasting forever.
God's love is foreshadowed in the burning bush that appeared before Moses: the bush was afire but was not consumed. So it is with God's love. He who is love dwells within us and through us. If we accept Him and live in congruence with Him, it is heaven. If not, it is Hell. It is the same "fire" of divine Love; it is the same God; the only difference is in how we respond to Him.
On the Day of Judgement, we shall receive exactly what we have chosen by the choices of our lifetimes. It is His gift, but our decision. In other words, there are people who are living in heaven, or who are living in hell, right now: in their hearts, by the choices they are making. The key is that in this life the choice can still be made, we can still repent.
the unworthy,
-- John