Oh, the seasons—those shifting annual weather patterns that all you Californians have forgotten about. Being an east-coaster, Cummings saw plenty of seasons come and go and seemed to be totally obsessed with them. A bunch of his poems make use of them as symbols, with spring being one of his faves. In "my father moved through dooms of love," the speaker seems to use the seasons to symbolically track the stages of his father's life, from the youth of spring to the death of winter.
Lines 9-11: We get our first feel of the brush of the seasons when we hear about the speaker's Father's "april touch," which has the amazing ability to "[drive] sleeping selves to swarm their fates" (sounds impressive). This mention of a spring month makes us think that the speaker is remembering his father when he was young. It also seems to be playing off the idea of spring as a time of renewal and new growth. The father doesn't use his green thumb in a literal garden, though. He uses it to inspire people to grow as human beings to take their lives into their own hands.
Line 37: Next, we move into fall in the "septembering arms" of the speaker's dad. Cummings is pulling some of his Cummings-y tricks here, by taking a proper noun, which is usually capitalized, and transforming it into an uncapitalized adjective. When the speaker uses the autumnal month of September to describe his father's arms, we really get a sense of the way time is beginning to take a toll on his dear old dad. Just like the leaves that start to turn and the flowers that droop, the speaker's father is entering the final phase of his life.
Lines 41-42: We get another mention of an autumnal month with the lines "proudly and(by octobering flame/ beckoned)as earth will downward climb." Notice that Cummings pulls the same Cummings-y trick here that he did with September, by turning the month of October into an adjective. In this case it describes a fire that seems to be drawing the father closer and closer. Since October is later in the year than September, we're going to assume that the father is getting older, and the "flame" that he's being "beckoned" toward is inevitable death, and/or the irresistible process of aging.
Lines 47-48: Winter doesn't get mentioned by name, but it's not-so-subtly referenced when the speaker says of his father "if every friend became his foe / he'd laugh and build a world with snow." So, if we're following this whole seasons = the cycles of life thing, then this winter imagery is telling us that the father is getting older and older. What's cool about this guy is that he doesn't let it get him down. The speaker uses an image of somebody playing in the snow, which we tend to associate with children. So, it seems like speaker is telling us that his father was youthful right up till the end and made the best of his final years.
Lines 50-52: The speaker throws us for a loop by not ending the seasons motif with winter; instead he caps it off by bringing back spring. He describes his father as "singing each new leaf out of each tree" and says "(and every child was sure that spring / danced when she heard my father sing)." This confirms our suspicions that we were having with the playing in the snow imagery before. Even as he approached death, the father's focus was on life and inspiring those around him.