Most of the best LEGO costumes I've seen have followed a similar method for making the head: a base and a top both made from styrofoam, with a strip of card folding around the diameter sandwiched between them. If that's not the best explanation, than see for yourself: two of my personal favourites can be found here and here.
Just about every costume I've done so far has prominently featured wire. Wire is great because it's almost limitless in what it can be shaped to. However, the method discussed above made a lot of sense. In previous costumes, wire was a good option because the head shapes were intricate. The head of a LEGO minifigure is quite straight forward, making styrofoam more appealing; the only work one would have to do is taper the edges into smooth curves. Furthermore, the infamous head is so smooth and free of imperfections I felt that it would be impossible to replicate that smoothness using the wire method. As well as it has served me, a smooth finish it does not always make.
With this in mind, I decided to opt for styrofoam. I had enough to do the neck, which was straight forward and turned out as well as I'd hoped (see here). After this, I only had enough foam to make a start on the base; however, a start was all I could make. Days after making this start, I began to resent the difficulties I appeared to be having in sourcing more styrofoam for the project. In the end, I felt that I had to start thinking about contingencies. With this in mind, I began to experiment with a wire frame.
This has so far proven to be quite fruitful, to the extent that I have abandoned my search for styrofoam. The smoothness issue I feared was addressed by overlaying the wire frame with very light cardboard (the flexible type that cereals are typically packaged in). Despite substituting styrofoam for wire, I have followed the base/top method outlined above. Here is a quick pictorial rundown of the progress so far (written instructions will eventually follow):