I was wondering if the length of the carbon backbone affects hydrophobicity of amino acids (i.e. longer carbon backbone of side group = less polar = more hydrophobic).
Your assumption that hydrophobicity of amino acids increases with length of the carbon chain is correct. Also, a branched chain can have a lower hydrophobicity than a linear chain with the same number of carbons, because it has a decreased exposed surface area.
When polar acidic/basic functional groups are introduced, the side chain becomes more hydrophilic. For example, phenylalanine is classified as non-polar because of the hydrophobic benzyl side chain. Attaching a hydroxy group in para-position in the phenyl ring leads to tyrosine, which is more polar. This is due to its acidic phenol group which is deprotonated at physiological pH. Note that in both cases, the number and arrangement of carbon atoms in the side chain is the same.