Let $C$ be a nice (smooth, absolutely irreducible, etc...) projective algebraic curve given by $$ C: F(x,y,z)=0. $$ Suppose we have a point $(x_0:y_0:z_0)$ over $\mathbb{F}_p$ for a prime $p$ of good reduction.
Question: How one can see if this point over the finite field lifts to a point in $\mathbb{Q}_p$ (p-adic numbers) and how to obtain that point?
I know that there is Hensel's lemma, but I saw it only for lifting roots of polynomials in a single variable.
Actually, you can use whatever version of Hensel you know for doing this directly. I’ll answer your question for an affine curve $f(x,y)=0$, and leave it to you to make the argument projective.
You know that as elements of $\Bbb F_p$, either $f_x(x_0,y_0)$ or $f_y(x_0,y_0)$ is nonzero. Say that $f_x(x_0,y_0)\ne0$ — just choose any $b\in\Bbb Z_p$ such that $\widetilde b=y_0$, and now you want to solve $f(x,b)=0$ for a value of $x$ that has $\widetilde x=x_0$. Now use Hensel.