The main purpose would be to simply be poetic, and use an interesting metaphor for the fog. Cats are known to be quiet, stealthy, adept, and able to scale and inhabit places that other creatures can't. So, comparing the fog to a cat is an apt description because fog too is silent, stealthy, and creeps into places that others can't. Also, in this beginning part of the poem, Eliot is describing a run-down part of the city; he mentions the "half-deserted streets", the "one-night cheap hotels," the "sawdust restaurants with oyster shells." He emphasizes the run-down atmosphere by using the word "yellow" repeatedly; this is no beautiful fog that creeps beautifully before a glowing moon. Describing it as yellow emphasizes the industrial, polluted nature of the fog. Most likely, in the city, the pollution from industrialization is so dense that it turns the fog yellow. That this yellow, dirty color brushes against the windowpanes indicates that it is right at their door, and hard to escape. Eliot was a modernist writer, and modernists tended to bemoan the dehumanization and pollution of the industrial age, so Eliot's mentioning of the "yellow" fog plays into that.
Also, the part of the city that he is describing is rather decrepit, and with that image comes the fact that there are probably many stray cats that are running around in the alleys. So, as the fog creeps through this part of the city, alley-cats are probably fairly common and a part of that run-down atmosphere. So, rather than comparing the fog to say, a snake, Eliot picks an animal that is regional and logical for that part of town.
I hope that those thoughts help a bit; good luck!
The description of the fog as a cat is the use of personification. Eliot gives the fog the qualities of a living creature in order to give it more importance. The fog underlines the feeling of loneliness established in the first section of the poem, also suggested by the "one night cheap hotels" and the "sawdust restaurants". The description mirrors the loneliness of the Prufrock, who is searching for some meaningful human connection. The sleepy, cat-like fog is something tangible that Prufrock identifies with in his own search. He says there will be "time" for the fog, just as there will be "time" for him to "meet the faces that you meet."
Personifying the fog makes it easier to show Prufrock's identification with it - one living creature to another. The description of the fog, combined with the repetition of certain words and the rhyme scheme, heighten the song-like qualities of the poem. Eliot labels this a "love song" and uses these elements to the romantic introspection of the speaker, Prufrock. This is not a love song in the traditional sense, of course. This poem is one of the first modern poems, and Eliot mixes these traditional romantic elements with the modern images of urban loneliness and social anxiety to challenge the traditional. Prufrock is an anti-hero, an awkward and unsure protagonist in the story of his own life.
In "The Love Song of J Alfred Prufrock", TS Eliot tries to portray the unhealthy and perverted city life using the symbols of cat & fog. This perversion is of both sorts- Inner (or moral) & sexual. In addition to that, when the adjective YELLOW is uttered, a sick, Jaundice-like mood gets created. Thus, style supports the theme.
And what's the use of the personification, the question arises. In his dramatic monologue, Eliot has superbly used personification to make the poem obscure stylistically as well as thematically. Definitely, Eliot is a leading Modern poet; Modern age is itself complex and obscure thereby. And according to Eliot, a poet's writings should necessarily bear the marks of his Age. Probably that's why he's so frequent in using personal symbols.
Such is my opinion. It might help you.
The fog is a reference to WW1 mustard gas - he is describing how close the first world war came to destroying the world and how many men were lost. The other fog references here refer more to Robert Frost. Note the timing of this poem. It is a mustard gas reference.