Paul Kaye on his journey from Dennis Pennis to the National Theatre

The temptation to open an interview with Paul Kaye with the question 'so how come you're not funny anymore?' – as he once infamously asked Steve Martin during his brief stint as Dennis Pennis – is overpowering. But it's also oddly appropriate, considering Kaye has reinvented himself in recent years as a 'straight' actor.

So I get it out the way. He laughs, but with the tone of a man who's heard the line a thousand times before (and, let's be honest, he probably has). "Pennis was in part driven by an intense feeling of failure and disappointment and a sense of having nothing to lose; I see him less as the beginning of my career and more as the beginning of the end of my adolescence."

If Pennis was the end of adolescence then his latest role, as troubled Midshipman Harry Brewer in the National Theatre's revival of Timberlake Wertenbaker's Our Country's Good, surely marks his ascent to middle-aged maturity.

Brewer, says Kaye, has "all sorts of issues raging within him". In a play with no shortage of pain – it chronicles the arrival in 18th century Australia (then New South Wales) of a group of convicts and sailors who find solace in staging a production of The Recruiting Officer – he's perhaps the most tortured character of all.

He researched the real man behind Brewer and discovered that he came to the navy late, around 40, having previously worked as a clerk. At a time when most Midshipmen were upper class, he was one of a small percentage who'd climbed up from the bottom rung of society.

"His trauma is woven into the whole piece – he's riddled with the greater damage of everything. Spiritually, mentally, physically, he's dismantling before our eyes and there's a deep sadness to that."

Directed in the Olivier Theatre by Nadia Fall, the production features original music by Cerys Matthews, best known as lead singer of Welsh band Catatonia. This chimes with Kaye, whose colourful CV includes a spell with a punk band. "It's wonderful to have her involved – she knows her stuff."

Wertenbaker, he adds, was very open to discussing new possibilities for the play in rehearsals. This is perhaps surprising considering the fact her 1988 text has achieved modern classic status and is a staple of school drama (I was in a production myself, many moons ago). He reveals there's "a lot of singing" in the production, though "it's not a musical".

"I love the fact that writers still discover stuff in their own work – it's a beautiful thing," he enthuses.

Mention of music brings to mind Kaye's work in Matilda, the RSC's runaway hit show in which he originated the role of the heroine's cartoonishly awful dad. "I had no idea of the journey that show would take me on, I had no sense of the adventure and the challenge of it all."

And, looking back on it now, where does it rank? "It was the highlight of my career by a mile," Kaye replies, between chomps on a sandwich in the National's press office. "To be part of a production of that calibre, it was…" he tails off and takes another bite.

'Scruffy chancer'

Kaye's move into performing was a gradual one. He initially studied theatre design at Trent Polytechnic, under an inspiring tutor called Malcolm Griffiths (an old-style "corridor teacher"), who left Kaye believing "I could take on the world, I just wasn't sure how".

Following this he set up a theatre company ("we got some great stationary and then split up") before a mix of playing in bands and graphic design jobs, including a spell "behind enemy lines" working on merchandise for Tottenham Hotspur (Kaye's a die-hard Arsenal fan and is today sporting one of their more lurid 1980s away shirts).

"Priority number one was to be in a band, and that didn't happen," he reveals. "But if you're creative and are looking to express something you can transfer that into writing, acting, comedy."

Kaye's breakthrough as Pennis on The Sunday Show in the mid-90s undoubtedly closed a few doors (he's unlikely to work with Hugh Grant) but it opened many more, and his acting career has been pretty consistent since. His plethora of film and TV credits range from Woody Allen's Match Point to recent hit Game of Thrones, in which his character (Thoros of Myr) has achieved the notable distinction of still being alive.

But what threads it all together? "When I look at what I've done there's been a core that is consistent, it's just been expressed in different ways. Admittedly it's been pretty random and chaotic." He says this with a grin – the air of the school clown that got lucky and knows it (Jonathan Ross once described Kaye as a "scruffy chancer", which seems about right).

So after music, design, comedy and acting, what could possibly be next? "I still think I've got an album in me – and there's other stuff I want to write. But I think I'm in a good place right now. I've somehow arrived here at the National and it feels like the most exciting challenge of my life. I couldn't ask for more than that."

Our Country's Good runs in rep in the NT Olivier until 17 October 2015