As @Jim mentioned in his comments, Philip is trying imagine what his life would have been like if he were Watson.
Going a bit further, the text you referenced suggests that although Philip hates Watson, he is envious of him and wonders what his life would be like if he were Watson instead.
"to throw himself into the other's skin"
the author is making use of a literary device known as imagery
In literature, one of the strongest devices is imagery wherein the author uses words and phrases to create “mental images” for the reader. Imagery helps the reader to visualize more realistically the author’s writings. The usage of metaphors, allusions, descriptive words and similes amongst other literary forms in order to “tickle” and awaken the readers’ sensory perceptions is referred to as imagery. Imagery is not limited to only visual sensations, but also refers to igniting kinesthetic, olfactory, tactile, gustatory, thermal and auditory sensations as well.
The gushing brook stole its way down the lush green mountains, dotted with tiny flowers in a riot of colors and trees coming alive with gaily chirping birds.