(Excel) Conditional Formatting based on Adjacent Cell Value

You need to take out the $ signs before the row numbers in the formula....and the row number used in the formula should correspond to the first row of data, so if you are applying this to the ("applies to") range $B$2:$B$5 it must be this formula

=$B2>$C2

by using that "relative" version rather than your "absolute" one Excel (implicitly) adjusts the formula for each row in the range, as if you were copying the formula down