Dispersion forces - what determines which molecule will have its electron cloud repelled? Here is an image I found which illustrates dispersion forces:



One helium has its electron cloud repelled by the other helium's electron cloud, which exposes the nucleus and gives an induced dipole which holds the atoms together weakly. But when two helium atoms approach each other, what determines which atom will have its electron cloud repelled? I would think that both electron clouds would equally repel each other. Also I am guessing that this effect will get stronger as temperature is lowered (since helium is a liquid at very low temperatures), but I am not sure why.
You are correct that the electron clouds would repel each other equally.

The induction as shown in your second picture would require another outside entity, polarizing one of the He atoms, which would then affect the other.

At an arbitrarily show distance, we might describe a molecular orbital formation between the two helium nuclei, as shown here:


(source: mhhe.com)  

In this case, without the external perubation, the orbital shape (and hence the electron clouds) is (are) quite symmetrically distorted.

However, it becomes nonesense to describe a system of two atoms.  In liquid helium there are manifold other charge fields distorting these nice shapes, and you may get some asymmetry as shown in your picture above.