What are the biological mechanisms behind the increase in cancer risk and alcohol consumption?
There aren't many studies which associate between alcohol consumption and cancer to my knowledge. The association is more with metabolic diseases because the liver is affected.

Alcohol is oxidised to acetaldehyde by alcohol dehydrogenase. Acetaldehyde may also, in turn affect the cells by DNA damage if it is not metabolized quickly to acetyl-CoA (perhaps in a manner similar to formaldehyde or by formation of Schiff's base with nucleotides). The other way by which alcohol is metabolised is through cytochrome-p450 (CYP2E1) which produces ROS (Hydrogen Peroxide), that may lead to carcinogenesis.

You might want to check this review.