At a bare minimum, you need a power supply, and an AtMega or AtTiny chip.
The AtMega and AtTiny's have an internal clock, 1mhz. It's not terribly accurate for timekeeping purposes (1% error, I think, as opposed to 20ppm = 0.02% for the external clock), but I see you have an RTC, so a 1% drift is probably fine (I can't see your code, so I don't know if the internal time is used for anything). Remember that you will need to adjust your code for the different clock rate, anything that uses anything time-related will run at 1/16th speed (delay, serial comms, etc). You will need to set the fuses to use the internal clock.
If your power source isn't 5v, then you need a transformer.
What type of AtTiny or AtMega you use depends on your code, and how many pins you use. I see 5 pins, they are all used as digital I think? You might get away with one of the 14-pin AtTinys. I suspect your code is probably pretty small. You can probably get away with nearly any AtTiny. To select an AtTiny, go to http://www.atmel.com/products/microcontrollers/avr/tinyavr.aspx
Nick Gammon (who will probably be providing his own answer as I type) has an excellent site on how to put the parts together, at http://www.gammon.com.au/breadboard
This site mentions programming the bootloader. On a low-memory chip, you do not need a bootloader, you can upload your program using SPI (same way as you upload the bootloader in Nick's example). Alternatively, you can use a programmer like the USBASP, which makes the whole process a lot easier. When you build your final board, I strongly suggest putting a 6-pin ICSP header (same as the 2x3 pins on the end of your Arduino board) on, to make reprogramming easier. You plug the USBASP into the 6-pin header, and click File->Upload using programmer.
If you want your project "really nice", you can design and etch your own PCB. If "good enough is good enough" (which for me it always is), I suggest you use stripboard (also known by it's brandname, veroboard).
In either case, spend some time thinking about your exact layout. Use a computer layout tool, and spend some time working out the best layout. I generally spend at least 5 times as much time on the layout, as I do on building the board - if a project is twice as big, I will spend twice as much time soldering, but at least 4 times as long laying it out.