Well, this speaker sounds pretty darn dissatisfied with his life. This very first line of the poem lets us know that the speaker's feelings of boredom are global; though he goes on to provide us with specific examples of stuff that bores him, this line tells us that he really thinks everything is a yawnfest.
'Ever to confess you're bored means you have no
Inner Resources.' (5-7)
In these lines, the speaker is quoting dear old Mom. Notice that the speaker's mom here doesn't say you should never be bored. She just says it's bad to confess that you're bored. So, it's not that the speaker is weird or weak because he feels bored with his life. The mom here seems to be saying that a sense of boredom or dissatisfaction is to be expected, and we have to rely on our "inner resources" to combat that boredom. Oh, and we definitely shouldn't admit to feeling it.
[…] I am heavy bored. Peoples bore me, literature bores me, especially great literature, Henry bores me, with his plights & gripes[…]
And the tranquil hills, & gin, look like a drag (8-11, 14)
We get it, Henry. You're bored. Enough already. In these four lines alone, Berryman uses four variations of "bore." And three of those "bores" occur in exactly the same spot in the line: in the second word. Berryman also starts and ends line 10 with the same word, "literature." He's really hitting us over the head here. He wants us to experience what the speaker is feeling: it's all the same, nothing is new or exciting. Uh. Mission accomplished.