This is what I've done:
$$X(x,y) = -y\frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$$
Consider the chart $(U,\varphi)$ , where: $$U=\mathbb{S}^1 - N$$ $$\varphi (x,y)=\frac{x}{1-y} \equiv u$$. Then:
$$\frac{\partial}{\partial x} = \frac{\partial u}{\partial x}\frac{\partial }{\partial u} = \frac{1}{1-y} \frac{\partial}{\partial u}$$
$$\frac{\partial}{\partial y} = \frac{\partial u}{\partial y}\frac{\partial }{\partial u} = \frac{x}{(1-y)^2} \frac{\partial}{\partial u}$$
So:
$$X = \frac{-y}{1-y} \frac{\partial}{\partial u} + \frac{x^2}{(1-y)^2} \frac{\partial}{\partial u}=\frac{-y+y^2+x^2}{(1-y)^2} \frac{\partial}{\partial u}=\frac{-1}{1-y} \frac{\partial}{\partial u}$$
Where I have used that $x^2+y^2=1$.
However, I have not been able to express $X$ exclusively in terms of $u$.