I have that
$$ \phi(r) = \cases{ A_1\frac{e^{-\kappa r}}{r} &\text{for } r>a \\ \frac{A_2}{r} +B_2 &\text{for } 0< r< a } $$
where $A_2 = \dfrac{q_1}{4\pi \epsilon_0 \epsilon_r}$
I need that both $\phi(r)$ and $\frac{d \phi(r)}{dr}$ are continuous at $r=a$, in order to determine the constants $A_1$ and $B_2$. How set up these boundary conditions?
I believe this is the correct way is to set
$$ \phi_{r>a}(a) = \phi_{0< r< a}(a) $$
and
$$ \frac{d}{dr}\phi_{r>a}(a) = \frac{d}{dr} \phi_{0< r< a}(a) $$
Is this correct?
A function $\phi$ is continuous if $\lim_{r\to a^-}\phi(r)= \lim_{r\to a^+}\phi(r) = \phi(a).$
So what you have is almost correct, you just need to clarify things a bit more, for example: since $\phi$ is clearly continuous on $(a,\infty)$,
$$\lim_{r\to a^+}\phi(r) = \lim_{r\to a^+}\phi_{r>a}(r) = \phi_{r>a}(a).$$