I have been going online looking for some simple 555 timer circuits, just to verify my thought process, and they all relatively looks the same... but whenever I attempt to power my circuit, the bjt or the mosfet (I switch between the two to compare results) overheats. I'm not concerned with efficiciency currently, I'm just interested in proof that the concept works. My coil is ~10uH

I have used the exact circuit seen in the attached image, except the only difference is my bjt is a pn2222 & my power supply is 5V.

 

This is an example of a "bait" schematic from the Internet.

While it's conceptually sort-or-correct, it's not a good design.

There is no path for energy stored in the inductor to flow when the transistor shuts off, except to over-voltage the transistor into destructive avalanche breakdown.

 

Since you are apparently experimentin with the '555 you would probably be much metter off if you forget thus circuit and move in to the next.

 

Sensacell, pretty much has the right of it here, just for the sake of it though, your inductor is also on the small side so you would have to make sure you were operating at rather a high frequency to simply prevent a short circuit.

 

Simulation shows that, for the R and C values shown in post #1, the pulse frequency is about 20kHz and the BJT would be dissipating several Watts on average, if it survived the excessive collector current and the high voltage spikes generated by the inductor's back-emf. A BD139 would be operating well beyond its rated current and power.

 

So I was originally testing the circuit out, and decided to think it through this morning. I figured a cap would be needed for the feedback from the inductor, but I thought (maybe they know something I don't)... clearly that was a mistake. So I did a few simulations this morning, and this looks like it may work.

My biggest question is, would this actually transmit the power? I haven't done any work with wireless circuits in a long time, and I'm trying to go back and learn through practice. Thanks for all your help!
(the pulse is simulating the 555 timer).

 

would this actually transmit the power?

Well, it will transmit some power. How much could be usefully collected would depend greatly on the receiving arrangement.

 

For a 10μh coil, the current will increase by 0.5A/μs with a 5V source when the transistor is ON, so you need to limit the ON-time of the transistor to limit the current to the desired maximum.
Thus if you want a 5A maximum, the 555 output should be high no more than 10μs with sufficient OFF time for the current to decay to zero (from the added free-wheel diode across the coil) before the next cycle.