Larkin's poems are often laced with irony, and "Mr Bleaney" is no exception: a series of ironic moments give the poem its dark humor.
For example, in lines 5-7, this juxtaposition of image and dialogue is jarringly ironic:
[...] Whose window shows a strip of building land,
Tussocky, littered. ‘Mr Bleaney took
My bit of garden properly in hand.’
Maybe the landlady is kidding herself; maybe her garden is dead now that Mr. Bleaney isn't there to take care of it; maybe her garden is dwarfed by the rest of the ugly view. Regardless, as the speaker looks out at grass clumps and litter, the landlady's praise of the view contradicts the reader's expectations. (If anything, you'd expect her to make an excuse for it, or not mention it at all.)
Lines 7-9 contain a bit of situational irony:
Bed, upright chair, sixty-watt bulb, no hook
Behind the door, no room for books or bags —
‘I’ll take it.’
After the speaker has listed all the unattractive features of the room, he does what you'd least expect: rents the place anyway.
There's also a touch of dramatic irony in the detail about Bleaney's gambling habit ("He kept on plugging at the four aways"). Bleaney's bets couldn't have paid off much: he clearly never won enough to escape this awful housing! Along with the speaker, the reader knows that Bleaney's hopes for a lucky break were in vain.
Finally, the idea of Bleaney "Telling himself that this was home" carries an ironic charge. The reader understands that, if Bleaney really did tell himself this, he was sadly mistaken (at least from the speaker's perspective)—and in the speaker's imagination, he felt "dread" even as he told himself this. This lonely, bare space isn't a "home" in any true sense of the word.