Not sure this is what you want. I'd first solve for x, then use Series to compute the Taylor series for the function x(b) with parameters y and c. Note Solve returns three functions so would have three series. Below, I just compute the first five terms of the "normal" form for the first one at b=0

myx = x /. Solve[x^3 + b y + c == 0, x]

Normal[Series[myx[[1]], {b, 0, 5}]]