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)
- 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.
- Creating resonance with radio stations to milk them for energy.
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.
The last form is also pretty unlikely. It does work but Radio stations notice this pretty quickly (they monitor their electrical usage very closely) and they would call a car from the FCC real soon to find the guy who sucks all their power. The process would make it very easy to locate the T1k, because it would basically turn him into an antenna which sends a very strong signal. You can see this effect today when you carry a fluorescent lamp to a radio antenna which send on the MW band: It will eventually start to glow without any wires. Radio stations hate it when you do that.
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.