Philip Larkin was a "pessimist" in several respects. First, he did not believe in an after-life. He believed that death was final and that religion was a somewhat delusive human invention designed to help people cope with this brutal fact. Secondly, Larkin was a "pessimist" in the sense that he seemed to believe that most of what he valued in life, such as love and beauty, is mutable and cannot last.
Perhaps the classic expression of Larkin's bleak view of death appears in his poem "Aubade":
. . . this is what we fear -- no sight, no sound,
No touch or taste or smell, nothing to think with,
Nothing to love or link with,
The anaesthetic from which none come round. (27-30)
Death, the speaker says, is something
. . . that we can't escape,
Yet can't accept. (44-45)
Some might call this pessimism; Larkin himself might have preferred to call it realism -- a simple willingness to face facts, however unpleasant.
More evidence of Larkin's "pessimism" can be found in his poem "Church Going," in which the speaker assumes that religion will and almost certainly must fade and lose most of its influence. The speaker is left wondering,
When churches fall completely out of use,
What we shall turn them into . . . . (22-23)
Note the inevitability implied by the word "When": the speaker does not wonder if churches will become obsolete; he takes it for granted that they will. Likewise, he later assumes that even "superstition, like belief, must die" (34). Although the poem does end by suggesting that a few people, at least, will always seeking wisdom of some sort (59-63), in general the tone of the work might aptly be described as "pessimistic."
Similarly, in the poem "Talking in Bed," the speaker describes a now-strained romantic relationship in which
It becomes still more difficult to find
Words at once true and kind,
Or not untrue and not unkind. (10-12)
Here even love, which might have seemed at least a temporary answer to (or refuge from) nothingness, is mutable and fades.
It is not difficult to find many works by Larkin which seem to suggest a "pessimistic" view of life, such as the poem "High Windows." At the end of that work, the speaker thinks of "high windows":
The sun-comprehending glass,
And beyond it, the deep blue air, that shows
Nothing, and is nowhere, and is endless. (18-20)
Likewise, an emphasis on unappealing mutability appears in the final lines of "Sad Steps," where the speaker thinks of
. . . the strength and pain
Of being young; that it can't come again,
But is for others undiminished somewhere. (16-18)
Youth is a source of both "strength and pain," but it is inevitably mutable, even for those who "somewhere" presently enjoy it.
As to the second question, Larkin was indeed associated with the so-called "Movement" poets of the late 1950s. However, this was never a formal group with a defined or shared manifesto. Larkin himself was too much of an individualist ever to sign on for membership in any group that would dictate how and why he would or should write poetry. He sympathized with some of the principles associated with "the Movement," especially their desire to make poetry accessible again and their desire to avoid the excesses of both modernism and the sort of neo-Romanticism associated with Dylan Thomas. But in the final analysis Larkin was always his own man.