As an example consider C : y^2 = f(x) where f(x) = x^5 - 2x^4-7x^3 + 8x^2 + 12x = x (x + 1) (x - 3) (x + 2) (x - 2) over \mathbb{R}.
From this picture it is immediately clear that we cannot use the chords and tangents method to define a group law on the set of points of a hyperelliptic curve.