One note: The bird we call chickens probably didn't in exist pre-Colombian America. One report^1 contended that mitochondrial DNA in pre-1450 chicken remains from Chile indicated they had been brought over ~1350 from Polynesia. However, reanalysis^2 suggested that the carbon dates were skewed by oceanic carbon and the mDNA indicated Eurasian origins. This was corroborated by a very recent study^3.
Radiocarbon and DNA Evidence for a Pre-Columbian Introduction of Polynesian Chickens to Chile Alice A. Storey, et al. Proceedings of the National Academy of Sciences of the United States of America, Vol. 104, No. 25 (Jun. 19, 2007), pp. 10335-10339
Indo-European and Asian Origins for Chilean and Pacific Chickens Revealed by mtDNA Jaime Gongora, et al Proceedings of the National Academy of Sciences of the United States of America, Vol. 105, No. 30 (Jul. 29, 2008), pp. 10308-10313
Thomson, Vicki A., et al "Using Ancient DNA to Study the Origins and Dispersal of Ancestral Polynesian Chickens across the Pacific.(ANTHROPOLOGY)(Author Abstract)." Proceedings of the National Academy of Sciences of the United States 111, no. 13 (2014): 4826.
Genetic studies show that there were multiple independent domestication events for chicken across India, southeast Asia and China, all roughly 6,000 - 8,000 years ago. It was domesticated from the red junglefowl, which still exists today across that whole region, and which breeds freely with domesticated chicken.
Chicken raised today across the rest of the world (Europe, Africa and the Americas) belong to the Indian strain, that is, they are derived from the Indian domestication event. They may have spread to Europe via the middle east through the Indus Valley Civilization trade.
They were brought to the Americas by the European colonists.
Sources:
Multiple maternal origins of chickens: Out of the Asian jungles. Liu et al. Molecular Phylogenetics and Evolution, Vol 38 (1), 2006.
Overview of chicken taxonomy and domestication. Al-Nasser et al. World's Poultry Science Journal, Vol 63 (2), 2007.
Documenting domestication: the intersection of genetics and archaeology. Zeder et al. Trends in Genetics, Vol 22 (3), 2006.