I've been thinking of making a coilgun myself. I have 21pcs of 400V, 330uF capacitors I got for free a few years back.
There are probably a lot of sites dedicated to coilguns, but it would be fun to go through everything ourselves instead of just copying others.
Where to get supplied power
I've thought of mains, for obvious reasons. I have 230V in my country. Rectifying 230V, with thus 115V peak half-wave, would give (with formula 2*max_half_wave_voltage/pi) 2*115/3.14159 = 73V. With 110V this would be only 35V. (Please correct me if I'm wrong in these.) This ripply DC would then need to be filtered possibly with a capacitor to make it more even, I don't know how capacitors behave when charged with ripply DC.
A voltage doubler circuit could possibly be used to get higher DC voltage, like http://en.wikipedia.org/wiki/Cockcroft-Walton_generator.
In our case, we would probably like to get near the max rating for the capacitors.
Calculating output power
I assume I get the capacitors max rated 400V DC somehow. So, with the formula for energy in capacitors, (1/2) * C * V^2, I use C = 0.00033 and V = 400, I get 26.4 Joules per capacitor, so my 21 capacitors in parallel would then (by simple addition which seems logical here) give 554 Joules. Calculating actual kinetic energy of projectile is probably not possible very accurately, without taking into consideration a ridiculous amount of factors, but at least it is of course significantly less than the power in the capacitors. But for comparison, the muzzle energy of a .22LR rifle is 135 joules according to Wikipedia, which is quite a lot less that the electric power in my capacitors. But not nearly all electric energy would get transfered into kinetic energy, of course. I know little about weapons, but a .22LR is a weak rifle I assume. Another example, a 5.56 × 45 mm rifle (high power rifle I assume) has 1550 J muzzle energy (energy of bullet as it leaves barrel). But if I get one third of power of a high powered sniper rifle, that's pretty cool...
But in your case, you have a camera capacitor.. Wikipedia says "The nominal capacitance (for a photoflash capacitor) is around 80-160 µF (microfarads) in most disposable cameras, with larger values in larger flash units.". Let's say you have 160uF, 330V (max rating for your capacitor) would give 8.7 J, energy of an air gun. With 73V from rectified 230V mains would give 0.9J, something like an AirSoft gun I think . With 35V from rectified 110V mains would give only 0.2J. But of course, not actual kinetic energy in accelerated projectile, that would be even less.
Getting max rated voltage to charge the capacitors would be important as we see in the formula, voltage is squared which causes energy to rise exponentially as voltage is increased.
This is something I know nothing about, unfortunately. I'm as curious as you are how, how to design the coil. Enamel wire can very probably be bought online. Unfortunately, I believe the coil design required the knowledge of an electrical engineer or someone with equivalent knowledge (wink, wink, smart people here ;-) ).
Other things to consider
Then I think I remember that I've read that diodes (rectification diodes, not signal diodes) should be used to prevent current passing back into the capacitors (is this because a coil induces current back into the wires from the magnetic field it just generated?)
Switching the power into the coil
I think a solid-state semiconductor device, like a thysistor/SCR, is good. A mechanical switch does probably ripple when switched (when the contact bounces back and forth), this may not be wanted in our case, and large capacitor powers could possibly weld the contacts.
Theoretically, all power should be used up when the projectile passes the half-way of the coil (at least this would make sense, to prevent deceleration of the projectile back toward the centre). If there would exist a solid-state switch that can be turned off with DC power (like a triac can with AC) that would be very good to prevent the projectile being sucked back. But I don't think such a device exists. The time before being closed could be controlled with a microcontroller that does the timing, and an adjustment screw for trial-and-error adjustments. An adjustment screw would allow for different adjustments for different projectiles.
Merged post follows:
Consecutive posts mergedSwitching (more)
I believe (possibly) one or many Power FET's could be used to switch the power into the coil. These can also turn the power off, unlike a thyristor.
If one was to take a thin layer of fiberglass sheet, roll it around a shaft of the desired inner diameter of the barrel, and then roll the "magnet wire" around it. After rolling the magnet wire, additional layers of fiberglass could be applied and finally treated with the fiberglass hardening resin. This way we have a non-magnetic barrel, and fiberglass has high strength which is good to prevent deformations when the coil is turned on.
Found easily on the net.
By making the barrel from fiberglass, and making some kind of attachment system, one could make a modular design that allows for barrels of different inner diameter, length and other properties.