I assume from your question you want the essential parts of an Arduino along with your circuit in a permanent form.

Here are my steps:

1. Build the entire circuit on a breadboard or two. Then you know it all works.
2. Transfer it to a permanent form of circuit, testing as you go.

Take it a step at a time, and it'll work out nicely.

1. Build the entire circuit on a breadboard or two.

Then you know it all works, and that you have all of the components ready to make your permanent circuit

This will involve making and programming part of your Arduino on the breadboard. I don't think you'll need the USB part, so it can be through-hole components which are relatively straightforward to begin with.

Here is a link to instructions which show how to make an Arduino without USB.

It can be programmed using your existing Arduino, you don't need a programmer. For example: http://arduino.cc/en/Tutorial/ArduinoToBreadboard

I'd make a sketch of the circuit schematic to ensure I understood what I had on the breadboard, and to guide the next step.

2. I'd recommend you transfer the whole thing to 'veroboard' (stripboard) rather than make a PCB.

Veroboard/stripboard will be much quicker and cheaper. You could easily spend 10x longer learning to use Eagle well enough to get a PCB made, than it would take to design and make the entire circuit on veroboard/stripboard.

You can do a design on squared paper, but there are some CAD programs to help if you google for veroboard CAD. I have never used them, I use paper and a soft pencil or a vector drawing package. A friend used PowerPoint because that's what he had to hand.

Here is an example of someone who has built an Arduino on veroboard/stripboard.

It shows the design he produced for veroboard/stripboard. Try to get a reasonable layout for your design before making it. This is where the soft pencil and eraser come in :-) Typically the first couple of attempts are too big or too small. Make it easy for yourself and get plenty of squared of graph paper :-)

This link shows what the process will look like.

You can follow veroboard/stripboard arduino design, and test that it works. Then focus on your extra circuit.

Use a socket for the microcontroller, and so don't solder it in directly. Most of the other parts are a few $'s total, so I'd get a few of each part for spares and practice (To put it in context, some electronics companies charge more for delivery than those parts will cost, so getting several sets of parts makes sense, especially if you intend to make some more things).

Total cost for the Arduino part should be under $10.

Good luck, and I hope you enjoy it.

Power seems to be an important aspect in your design, so you might consider using different LEDs. Low-current LEDs for instance take 2mA instead of 20mA.

Or maybe 5mA through your existing LEDs will produce enough light for your taste.

Another angle would be to design your LED drive circuit for as low a voltage as you can. If you can design it for 3V and you use a switched power supply to create this 3V from 5V (or maybe 12V), you have saved 40% energy. This is especially effective when you use a battry, which produces a lower voltage over time.

I guess that when you want to iluminate (nearly) all LEDs you can get by with illuminating them each a bit less. This logic could be incorporated in your software, so the worst case current would be reduced.

You must design your circuitry for the worst case, but for battery life it might be more realistic to calculate with averaged cases. So get some more info on what you want to show when you need to know how a given battery will last.

As often, your one good question results in an avelange of questions, and some questioning of your basic assumptions. That's system design :)