The process of introducing a phosphate group onto a serine residue at position 5 within the heptapeptide repeat (YSPTSPS) of the C-terminal domain of RNA polymerase II. Typically, phosphorylation of serine 5 (Ser5) occurs near the 5' ends of genes. It is generally still observed in the middle of genes, overlapping with phosphorylation of serine 2, but is generally not present at the 3' ends of genes. In vivo, Ser5 phosphorylation occurs primarily through the action of TFIIH (KIN28 in S. cerevisiae, CKD7 in metazoans).