#JELLY 10 bytes

    f⁴L
    ŒgÇ€»/

Explanation:

    f⁴L
    f⁴      -Filter by the character argument.
      L     -Return Length of filtered String.

    ŒgÇ€»/
    Œg      -Group string by runs of characters.
      ǀ    -Run above function on each group.
        »/  -Return the largest in the list.