Consider an elliptic curve defined over the field of rational numbers and given by $$\mathcal{E}_n: y^2=x^3-kx,\ k \ne 0$$
Let $B = \left(\dfrac{r}{s^2},\dfrac{t}{s^3}\right)$ with $r,s,t$ coprime. Can we find the constants that appear in the definition of a height function, i.e. can we find $C_1$ and $C_2$ in terms of $B$ and $k$ such that for any $A \in \mathcal{E}_k(\mathbb{Q})$ we have $$\left\{\begin{matrix} & h(A+B) &\le 2h(A) + C_1\\ & h(2A) &\ge 4h(A) - C_2 \end{matrix}\right.$$
Thanks for all the hints and any good reference on this subject would be appreciated!
I think this can be done. I suggest you read Siverman and Tate's book "Rational points on elliptic curves". In particular, this is done in general for an elliptic curve over $\mathbb{Q}$ in Chapter III, Sections 2 and 3. If you follow their arguments and replace the curve by the one you are interested in, you should be able to write the constants explicitly... but it will take a bit of work to go through the whole argument (p. 68-75) to get the constant right. However, it will be worth it! Because in the process, you will understand the proof of the bounds.