I have blogged already on the generalities of making a perfect cup of tea(1), and how much difference the quality of the water makes(2). This blog is specific to answering how to brew green tea so it does not taste bitter.
The water is too hot. If your water is straight off the boil, it will literally burn your tea leaves. Allow your water to cool for 1.5–2 minutes. This will bring it down to around 80-85 degrees Celsius | 180 degrees Fahrenheit, perfect for allowing the flavour to draw slowly.
You are not storing your tea properly. Tea should be stored in airtight containers away from light, heat and moisture (also strong smelling food). If not, your tea will become stale and bitter.
If you use a good quality green tea, and follow the above, you will be able to get three to five delightful, non-bitter brews ~ of subtly different depths and flavours ~ from the one pot.
And don’t forget the most important part ~ take the moment, sit back, breathe, and enjoy. Life is short, drink good tea. ~ GreenWitch Tea
Of further interest~ brewing tea in different cultures