If a positive charge A moves towards another stationary positive charge B then the A's electric potential energy increases. But shouldn't the electric potential energy of B also increase as it is also in a way moving towards the A inside the A's electric field? So shouldn't the total increase in electric potential of the system be the double of the increase in any one of the charges?
1) The potential energy might decrease if the charges A and B are opposite. 2) Assuming A and B are of like charge, the increase in potential energy is held in the configuration of the system A and B together. The potential energy is not attributable to either A or B independently. This is also true for the gravitational potential energy. However, in the case of two objects of disproportionately different sizes, it is often convenient to neglect the motion of one of the objects (the larger one, since it accelerates so little). And in this case, one often talks about "the potential energy of the smaller object" since that's the only motion we care about, when in fact the potential energy is contained in the configuration of the system.
The potential energy of the system (of charges A and B) in its final configuration equals the total work that some outside agent has to do, in order to bring both charges from infinity to their final locations. Start with both A and B at infinity. Bring charge A to its final location. This requires no work, because charge B is still infinitely far away. Bring charge B to its final location, a distance r_{AB} from A. This requires work kq_{A}q_{B}/r_{AB}. The total work and the potential energy are therefore 0 + kq_{A}q_{B}/r_{AB}.
But doesnt the potential of A and B increase as A or B gets closer because both of the particles a simultaneously changing their positions in the elctric field of each other?
There's just one potential. The potential due to the configuration of A and B. There's no two separate potentials that you are thinking of. There isn't a potential of A and a potential of B, there is just potential of A and B.