These answers were very helpful. Thank You!

This should help.
http://honest-food.net/2009/10/11/how-to-cure-green-olives/

Hank is the king when it comes to these technique questions. You can find both oil and water methods in the post I linked above.

If you use my method, don't bother to crack the olives.

Should I cut or crack the olives to open them to the brine?

I have seen that you should keep rinsing until water runs clear.

Here's a brine approach for green olives (6 mos total time)

http://www.nytimes.com/2007/10/17/dining/172orex.html?ref=dining

Curing olives is easy! I use a brine method which I have been doing for many years. First pick the olives and make sure they are clean and insect free. I use only green olives for this method as the black ones get too soft. Make the brine: in quart of water add enough salt so that an egg will float(start with 1/2 cup and keep adding until the egg floats). Put half the olives in a quart jar. Put in a quarter of a lemon slice (and a clove of garlic if you want). Add the rest of the olives. Pour brine to cover. Top with a thin layer of olive oil. Close the jar. I do not seal it. I put a baggie on top and clamp the lid. Put the jar in the refrigerator and forget about it for 6 months. When you open the jar, the oil will be solid and messy. Take out as many olives as you need and rinse them and sprinkle them with herbs and olive oil. Be sure to keep a thin layer of oil on the jar and keep refrigerated. This recipe is also posted on food52

I have both green olives and purple ripened olives. I was hoping to be able to cure them without lye. I have been soaking some of the green olives in water (after hitting them with a wood mallet to open them a bit), changing the water twice a day but they are still bitter after four weeks.

It is really a lot more work than it's worth to do at home, generally. Most olives are cured using lye, if I remember correctly. If you're really set on doing it at home, you could try a dry salt cure, which involves a LOT of rock salt and a fair bit of time. I have a friend with an olive farm in Egypt and my recollection is that he fills a barrel with layers of olives (this really only is done with ripe, ie. black, olives) and salt. He regularly drains the water off and turns the barrel. If you're really set on doing this, let me know and I'll get some suggestions from him on what you could try at home with only the fruit from one tree.

Check out the comments of that post as well for a non-lye method.

This is the best approach I could find.

http://blog.ruhlman.com/2009/10/how-to-cure-olives.html

However, note that using lye is very dangerous.

What kind of olives (color) do you have?