When Matthew Rhys first took his partner Keri Russell to west Wales, he insisted on visiting Laugharne to show her a writing shed used by Dylan Thomas. They also stopped at the poet’s favourite pub. She reminded him of these facts recently. “Oh my God,” he says. “I really am a cliche.”

The actors, who starred in the TV drama The Americans, will perform together again on Tuesday at 92NY, an arts and innovation centre in New York that was responsible for bringing Thomas to America in 1950, catapulting him to fame on a triumphant but ultimately tragic series of tours.

Rhys, who will take on the role of Thomas in a dramatic reading of Dear Mr Thomas: a New Play For Voices, helped persuade 92NY to commission the one-act play as part of its 150th anniversary celebrations. The work, by Welshman Christopher Monger, is an attempt to imagine the chaos of Thomas’s final years and the genesis of his best-known play, Under Milk Wood.

Central to the story is Thomas’s relationship with John Malcolm Brinnin, who, as the new director of 92NY’s Poetry Center in April 1949, was determined to bring Thomas to New York and offered him $500 to give readings of his poems.

“Brinnin was – especially at the beginning – so in awe of his talents, so hellbent on bringing him,” Rhys says by phone from New York. “But he became this enormous problem for Brinnin.

“Brinnin famously said, within the first 48 hours of Thomas landing in New York, he’d drunk more than he’d ever drunk, eaten less than he’s ever not eaten and stayed awake longer than he ever had in his whole life.”

The actor adds: “Brinnin was hit by this whirling dervish, and didn’t quite know how to manage him. He was trying to keep him on the straight and narrow, trying to make sure he didn’t spend his money like water. It was like trying to nail Jell-O to a wall.”

Thomas made his debut in the US in February 1950, “the first of those performances which were to bring to America a whole new conception of poetry reading,” as Brinnin would put it. The poet EE Cummings, who was in the audience that night, was so moved by what he heard that he walked the streets until morning.

Brinnin had opened a Pandora’s box that both blessed and cursed Thomas. Rhys says: “Jack Kerouac, all the beat poets, were obsessed by him. When he spoke at 92NY, he was about to finish without reciting a poem of his called Fern Hill, and the audience started to chant ‘Fern Hill! Fern Hill!’ It’s kind of staggering to think that a poet of his time would sell out this enormous space and have the audience chanting; that’s a ritual that’s only reserved for rock stars normally.

“He was the rock star poet and with that came great problems: the adoration, the fandom, the alcohol in excess, how it affected his writing. It was everything. He wanted it all. He was meeting Charlie Chaplin, being courted by Stravinsky to write opera. He was being given the keys to the kingdom and with that came incredible excess and pressure.”

Brinnin organised a reading tour, the first of four, that took Thomas to colleges and universities across America. One challenge was what to wear. He wrote to his parents: “In Chicago, it was bitterly snowing. In Florida, the temperature was 90[F]. And New York itself never has the same sort of weather two days running.”

Rhys says of the tour: “They ripped through it. He did 40 universities in three months from Nova Scotia all the way down to Florida, and all the big ones in between, and even the Museum and Modern Art and all sorts of places.

“The workrate was incredible. For as much as he is portrayed as the roaring boy, the wild man, he missed one performance and everyone said, regardless of how much he drank, his capacity to nail the reading the next day was always untouched.”

The premiere of Under Milk Wood on 14 May 1953 – Tuesday’s reading coincides with its 71st anniversary – was Thomas’s most memorable performance at 92NY. With glorious lyricism, the play chronicles a day in the life of Llareggub (“bugger all” spelled backward), a fictitious seaside village in Wales populated by characters such as Captain Cat, Polly Garter, and Mr and Mrs Pugh.

It is a classic of 20th century literature. Rhys sums up its appeal: “We are obsessed and disgusted and infatuated by each other – it’s that human interest. He’s basically saying what social media is now, this voyeurs’ element of how other people are obsessed with other people’s lives and how we pry into them are incredibly universal themes.

“Great humour courses through the whole thing but also you have that added thing which was his penmanship, his wordsmithery. His pace of writing was so slow because he used cut out the words individually and then move them around like a giant jigsaw puzzle because he said every word must have an equal amount of weight and meaning. If you have someone whose craftsmanship is so detailed it stands the test of time and proof of that is in endless productions we still see.”

Just six months after that first performance, Thomas died in St Vincent’s hospital in New York at the age of 39. A postmortem cited pneumonia, brain swelling and a fatty liver as causes, but alternative theories have been debated ever since.

Tuesday night’s cast includes Russell, Gopal Divan, Betsy Zajko, Taylor Trensch and Kate Burton, whose father Richard Burton popularised Under Milk Wood on BBC radio in 1954 and starred in the 1972 film adaption. Rhys comments: “She brings with her this wealth of history, where she has all these stories about her father, who was great friends with Thomas. When Thomas passed, he did so much care for Dylan’s family and for his name and for the longevity of his work.”

Rhys has played Thomas before, notably in the 2008 film The Edge of Love, co-starring Keira Knightley, Sienna Miller and Cillian Murphy, which offered a snapshot of the poet’s complicated love life during the second world war. He has also appeared in a production of Under Milk Wood in which the premise was that the narrator was Thomas himself.

But this one will be special. Although both appeared in last year’s film Cocaine Bear, it is the first time he has worked so closely with Russell since The Americans, in which they played two KGB spies in an arranged marriage who pose as Americans in the Washington suburbs. It ran for six seasons, long enough for the couple to establish a working shorthand. “Inevitably, if you live with anyone long enough, you can tell by a breath how they’re feeling,” he says.

More recently Rhys has been seen in A Beautiful Day in the Neighborhood and Perry Mason; Russell plays the US ambassador to Britain in The Diplomat. He adds: “She’s now at that point where when she does live things she gets very nervous – it’s like, ‘Oh, why have I let you talk me into this?’

“I did say, could you play Caitlin Macnamara [who was Thomas’s wife] and she’s like no, of course not! So she’s going to be giving Liz Reitell [Brinnin’s assistant and Thomas’s lover]. But it’s great. I’m very happy. We’ve always talked about working together at some point and I love the fact that something as personal as this will be our first time back together.”

• Dear Mr Thomas: A New Play For Voices will be performed at the 92Y on 14 May