There it is, right in the title. Dream. Berryman wanted to make sure we had the notion of dreams and the subconscious in mind when we read this poem (and close to 400 others with "dream" in the title). By titling this as a Dream Song, Berryman is kind of warning us that what follows may or may not be as it appears—just like in dreams.
with open eyes, he attends, blind. (10)
In the first stanza Henry is "sleepless," eyes wide open, and wicked sad. The second stanza begins with, "another thing he has in mind." Henry pays careful attention, "he attends" to it. But he is "blind." Here's a riddle: how can you be blind and have vision at the same time? Ding Ding Ding! That's right. When you're dreaming.
Those other things are in us, in our minds, even when we're awake. They may not surface, we might be "blind" to their existence when we are walking around going through the motions of our daily lives, but they are there. Think about it.
Often he reckons, in the dawn, them up. (17)
If the violence in the third stanza is dreamt or imagined as the speaker tells us it was ("But never did Henry, as he thought he did, / end anyone"), then the poem concludes at dawn with Henry exiting the dream world and re-entering reality. The trouble is, he's having a tough time making the transition. He doesn't just wake up and think to himself, "Wow, I don't think I'll be sharing that dream around the water cooler this morning." He actually has to mentally tally up all the people he knows to make sure that he didn't kill them in the night. Henry seems to be having an especially tough time making the transition from the subconscious to the conscious realm.