Hello friends, this is my take on a 5x5x5 LED cube, lets get started.

The first thing we need to do is get all of our supplies.
We're going to need...:
*125 LEDs (make sure they are red or yellow or the arduino will not be able to run it)
*wire, lots of wire
*5 3-8 line decoders
*5 NPN Transistors
*30 150 Ohm resistors
*a bread board ( preferably 3x 5 inches)
* soldering iron (always wear your safety glasses)
*sponge (for cleaning the soldering iron)
*conductive rods ( you can get 3/64 brass rods in a 6 pack at hobby lobby for about three dollars; get about 6 or7)
* wire cutters
*needle nose pliers
*wire strippers

Step 1: Step 1 Cutting the Rods

The rods at hobby lobby are one foot in length; however, we only need pieces that are 5 inches. we are going to build a frame for the cube then attach the LEDs to the skeleton.

First take your rods and get them all the right length. cutting two inches off every rod then cutting the rest in half to five inch pieces.
once you have all of them cut we can move on to the next step.

Step 2: Step 2 Make the Frame

once you have all of your pieces cut out take four of them ans put them in a square. make sure that they are at 90 degree angles or the cube will not fit together right. once you have them in place solder the corners. repeat this step four times until you have all five layers together.
Next we need to take three rods for each layer and make the ribs.
take a ruler and a permanent marker to measure out 1 inch increments on two opposite sides of each square. these points are going to be where you solder the ribs to the square, again make sure that they are at a 90 degree angle. you'll also want to measure out one inch increments on each rib, you'll thank me later.
when you're done you should have five squares with three lines running through them.

it should look like this

Step 3: Step 3 Columns

once you have the layers together the next step is to get the columns.
here's where the rest of the rods come in to play.
again using a ruler and a permanent marker measure out 1 inch segments on each rod.
solder LEDs to each point making sure to attach the longer anode (positive leg) to the rod so the shorter cathode
(negative leg) to the squares you just built. you'll need to make 25 of these

this is what each column should look like

Step 4: Step 4 Completing the Cube

Congratulations you're almost there :D
the best thing to do at this point is to take one of the squares and attach a column to each corner. make sure to attach the column to the square only by the cathode, if the rods touch it could short something out and that's not good at all.
After attaching the four corner columns you should then solder the next square to the LEDs above the ones that you just attached the first layer to. this will make the second layer of the cube. After you have done this to all five squares you will need to attach the rest of the columns moving from one side to the other, his makes it easier on you.
I found that the best way to attach the rest of the columns is to place the column through the bars and hold them in place with the needle nose pliers while using your other hand to solder the cathode to the corresponding layer.
once you have gotten all 25 columns in place we can move on to the next step.
here are a few pictures that might help you understand how to do it. (i know its a lot to do all at once)

Step 5: Step 5 the Breadboard

OK now comes the part where you're gonna have to look at a diagram.
I'm attaching two diagrams, the first is showing how to set up and wire the breadboard and the second is where each wire is connected to the cube. read each carefully and you shouldn't have any problems.

Step 6: Step 6 Programing (dun Dun Duuuuuun)

ok this is probably the easiest part. just plug in your arduino, then copy and paste this code in sketch and upload. ta da you're done. everything should run properly. i hope you had fun making your cube, good luck