Humans are electrical too
I'm a long way from an expert in interfacing biological and mechanical/electrical systems, but I'll give this answer a shot.
There's been recent developments in dissolving batteries which, with enough development, could provide your power source. Alternatively, have your scientists develop a synthetic method of power production similar to the way the human body does - releasing energy from glucose and oxygen. I'm a bit rusty on my biochemistry but I'm sure it's at least withing the realms of possibility.
If you look at prosthetics today, they don't connect to the body on a cellular level. I'm not sure it's wise for any implant to connect directly to cells, as cells have a habit of dying all the time. You'd want your implant to last longer than your average cell.
All an implant should need to connect to would be a structural component (likely a bone) and a control component (nerves). However, this will probably be different for each type of implant, and even for each instance of each implant due to the unique nature of people's bodies. In the case of something like an implanted eye, it would need to be inserted into the eye socket and attached to the muscles. Alternatively, you could remove the eye muscles and replace them with artificial ones attached directly to the skull.
To allow for high levels of flexibility when attaching implants you're probably going to need advanced 3D printing technology as well of keyhole surgery (and some absurdly strong local anaesthetic).
Nerve cells in both the brain(neurons) and the rest of the nervous system pass information through a mixture of chemical and electrical impulses. This is why it is possible to monitor brain activity using electrodes. With this in mind, it seems reasonable to me that sufficiently advanced medical knowledge would allow people to write software to interpret these electrical signals.
Of course, these signals are likely to differ wildly from person to person, so an implant would be a highly specialised and personal piece of equipment. One idea would be to separate implant systems into two parts, the implant itself and the part that interfaces directly with the body. This could allow for swapping body parts on the fly, for fun!
Things to consider
- The human body has a tendency of rejecting implants as foreign objects, often developing inflammation where it is attached and causing great illness. You could construct your implants out of non-rejectable handwavium or do something with sequencing the client's DNA, perhaps?
Sorry for slightly muddled answer, ask any questions in the comments and I'll edit to clarify. :)