What current states did Eastern Woodlands Native live in?

All the states east of the Mississippi River, plus those immediately west of the Mississippi. The easternmost portions of Oklahoma and Texas also count. Also southeastern Canada.

What was the weather / climate in these areas?

It's a big area, stretching from just south of Subarctic Canada to the subtropics of Florida, and from the Atlantic coast to the edge of the Great Plains. Not only does it depend exactly where you're looking, but also when. Going by today's standards wouldn't be too far off, for the most part, but there's also things like the Little Ice Age (during most of the colonial period) and the Great Drought (corresponding to the Medieval Climate Optimum in Europe) in recent centuries, as well as the various climatic shifts during the Archaic (after the Ice Age until 3000 years ago).

Where can I find a link to primary sources about the Eastern Woodlanders?

Is there anything in particular you're looking for? A particular polity or culture, maybe? Or a particular time period?