Let $C_{1}$ be a circle of unit radius. Let A and B be two points inside $C_{1}$. Now I want to construct another circle $C_{2}$ such that A and B lie on $C_{2}$ and $C_{2}$ is orthogonal to $C_{1}$ at their point of intersection(I want $C_{2}$ in such a way that it intersects $C_{1}$). I tried and failed to find a way to construct such a $C_{2}$.
Any help is appreciated.
On
Take the inverse of $A$ with respect to the circle $C_1$ (see here) to obtain the third point $C$. Now construct the circle containing $A,B,C$.
If I understand your question correctly, it is answered in Construction 2.1 of this paper: http://comp.uark.edu/~strauss/papers/hypcomp.pdf
Edit upon request: Given two points $A$ and $B$ in the interior of a circle $C_1$, invert $A$ through $C_1$ to $A'$. Now construct the circle through $A$, $B$, and $A'$. Sigur's answer contains a picture of this construction.
The paper discusses general compass-and-straightedge constructions in hyperbolic geometry, where, in particular, geodesics can be modeled as circles perpendicular to the unit circle in $\mathbb{C}$.