The speaker of the poem is also a poet, and throughout the poem he uses the extended metaphor of a fox emerging from a forest to describe the process of writing a poem. At the beginning of the poem, the speaker looks upon a forest at midnight and notices that "Something else is alive." The midnight forest is a metaphor for his own mind. It, like his mind, is dark and seemingly still. This reflects the speaker's mind before the idea for a poem has come to him. All he has, before something emerges from the forest, is "the clock's loneliness" and a "blank page."
The something else alive in the forest becomes a fox, which "Sets neat prints into the snow." The snow here represents the poet's mind, or alternatively the blank page before him, and the footprints of the fox represent the first impressions of an idea for a poem which appear in the speaker's mind.
At the end of the poem, the fox is described as having entered "the dark hole of the head," and the speaker says that, "The page is printed." In other words, at the end of the poem, the idea represented by the fox has become a poem, or at least something close to a poem. The speaker now has a poem to print upon the previously blank piece of paper before him.
When the fox first emerges from the forest of the speaker's mind, the reader and the speaker see only a twitching nose, and then two darting, searching eyes. Then the fox emerges, a body, and moves closer and closer towards the window that the speaker watches from. The fox eventually gets so close that the speaker can see the "widening, deepening greenness" of its eye, and smell the "sudden sharp hot stink of fox."
The fox, as noted above, metaphorically represents the idea for a poem emerging in he speaker's mind. This is alluded to in the title of the poem. Indeed, the fox is defined in the title as a thought, or perhaps the product of a thought. The fox's footprints represent the first impressions of a poem, meaning perhaps the idea, or a key image, or maybe an opening line. The vivid "greenness" of the eyes, and the suddenness of the fox's smell, then represent those aforementioned impressions becoming clearer, and more sharply focused in the speaker's mind.
It's significant of course that Ted Hughes chose a fox, rather than any other animal, to represent the emergence of an idea for a poem. First of all, the vivid red coat of the fox makes it a good choice, as red is a bold, primary color, and stands out clearly against the backdrop of white snow. This color contrast thus helps to emphasize how the idea for a poem can emerge vividly and distinctively. A fox is also a rather beautiful and graceful animal, and in this way also serves rather well as a metaphor for a poetical idea.
The fox, however, is really just a representation of an idea emerging in the speaker's mind, and so one might say that there is really only one character in the poem, represented in two different forms.