There will be few sights more incongruous in 2014 than Larry King, the doyen of American TV interviews, chewing the cud with Terrence "Pusha T" Thornton. It's an indication of how far the veteran Thornton has come up the rap rankings in recent years – and a sign that King has previously unheralded game.

These, then, are the good times for the Bronx-born, Virginian-raised rapper. He initially set the scene for what was to come alongside his brother Gene in Clipse. The duo's debut album, Lord Willin', produced by The Neptunes long before Pharrell Williams got Happy, remains one of hip-hop's most fascinating albums, full of bumping street rhymes and lean and mean beats.

Clipse are now on a lengthy hiatus (their last album was back in 2009, and there are no plans for a follow-up) so it's Thornton's solo work which has his name up in lights. His debut album My Name is My Name was released last year, a title he credits to a Marlo Stanfield quote from the Baltimore-set TV series The Wire.

There's a lot of glitter in the cast list – his new label boss Kanye West supervised the production, and there are guest performances from Williams, Rick Ross and Kendrick Lamar – but the grit in the grooves is all down to Thornton.

He’s as proud as punch of what he has achieved. “I give people what they want from me. People know me and they know I do street hip-hop. By now, you know what you’re getting with Pusha T because I’ve been in this game a long time.

“I’m only moved by a certain style of rap, and you have to really be a rapper’s rapper for me to like it. When I go to make something, it has to be a strong body of work, it’s got my references and influences and passions for hip-hop in it. It’s honest. A lot of people make good albums these days, but this is beyond that.”

Of course, he’s happy to credit what the people who were on board with him for the ride brought to the album. “Kendrick was just incredible, he’s got this magic, you know. Rick always delivers, always gives people what they want to hear.

“Kanye insisted that we make something unorthodox musically. He was uncompromising, he wanted me to write the hardest verses every time. Pharrell brings something that’s unmatched. He’s a real music guy and I’ve learned so much from watching him and listening to him and asking him questions.”

It must be quite an experience, then, to work with these producers who are very much at the top of their game. “They are what I call super-producers. I respect what they do. Unlike a writer, they can see the future. I don’t know what it is, but they always see something. You always have to take what they say and listen because they really give you gems.

“My time in the game has allowed me to work with certain people who under normal conditions you just don’t get to take your call. This is production at its best.”

Thornton had a very distinct environment in mind for the album. “It’s a car album, man. I listen to a lot of music in my house, but this is something to play in your car with the windows down. It’s articulate street rap and it’s got this amazing energy to it.”

Lyrically, his focus has never changed. “I speak from the perspective of the streets, that’s always been my thing. I try to show people my thought process and talk to them about what I see as being right or wrong. That applies to everything I talk about, from family to relationships. A lot of people just say things and I know a lot of people just think of rappers bragging, but I try to be honest.”

He's proudest of one track in particular from the album. "One of the most poignant songs I've ever done is Sorry Nigga I'm Trying to Come Home on the album. It's real and it speaks to people in the game. A lot of rappers will try to do that, but I think I'm one of the only people who knows about that for real. I still know about that experience because I know people who are in that world."

What he's found hardest of all about taking the solo route is having patience. He signed to the GOOD label in 2010 and, while there were a clatter of mixtapes such as Fear of God and Wrath of Caine, there was a long wait for the album to arrive.

“A lot of people who knew me from Clipse or the Re-Up Gang were expecting the album a long time ago, but it’s not as easy as that. I had to get new fans, I had to get used to people hearing me solo, I had to get used to being solo. A lot of people just wanted to hear me with my brother, but I think everyone has come around now. I treated it like I was new and put the same amount of work into being heard as I did at the very beginning.”

While it may have taken Thornton years to get around to releasing his solo album, he's confident that My Name Is My Name will stand the test of time. "That was the plan. I wanted to create something which would be around forever and I wanted to do that by not compromising. I don't think a lot of people come close to what I do and the energy I have on the album. They lack something, whereas I feel I have it all."