I know no canon answer but a bit about material technology.

Since the design of the T1k is decentralized, it probably doesn't have a central power supply. Just think when it turns into floor tiles to kill the guard - there is no blob left that stands out. The whole T1k flattens out.

So each cell cluster must have their own power supply. What could that be?

- Heat ([thermoelectric effect][1])
- Magnetism (walking through Earth's magnetic field)
- Chemical reactions like in fuel cells. He could filter chemicals from the atmosphere and combine them to generate heat or electricity.
- Billions of tiny capacitors which are pre-charged when the T1k is formed.

Note that the first three would not generate enough energy to let the T1k walk. Unlike humans, the T1k doesn't have a skeleton and joints, so walking will be a very expensive process since it has to mold itself.

So the T1k probably uses some form of high-energy capacitor or some magical form of energy production which Skynet developed while it built the time machine.


  [1]: https://en.wikipedia.org/wiki/Thermoelectric_effect