Former Vice President Joe Biden provided insights on his process for selecting a running mate and cabinet in an April 3, 2020 virtual fundraiser.  Pool reporter Sabrina Siddiqui of the Wall Street Journal provided the transcript:

"One of the things I learned a long time ago, and I really mean this, a good leader has to be willing to have people that are smarter than them, know more than they know about a subject, bring in people who in fact have an expertise you don’t have.
And so I am in the process and I actually had this discussion with Bernie. He’s a friend. We’re competitors. He’s a friend. I don’t want him to think I’m being presumptuous but you have to start now deciding who you’re going to have background checks done on as potential vice presidential candidates and it takes time.
So I called President Obama, not as to who but how soon you have to start that. Now the convention’s been moved back now another month so there’s more time now. It’s kind of presumptuous, but sometime in the middle of the month we’re going to announce a committee that’s going to be overseeing the vice presidential selection process. 
And I must tell you: There are a number of people like you and I’m not being a wise guy who have been helping me, they’re serious people who I’ve had discussions with about whether or not — not a cabinet position, because there’s nothing quid pro quo — but asking them are they willing to come into a government if I get elected? What are their circumstances? Who do they think I should be looking to? And there’s an enormous number of really qualified people.
And one of the ways to deal with age is to build a bench — to build a bench of younger, really qualified people who haven’t had the exposure that others have had but are fully capable of being the leaders of the next four, eight, 12, 16 years to run the country. But they’ve got to have an opportunity to rise up. They’re so many of them out there — women and men. So I’ve been talking privately with people like Tony [Blinken] and others, talking about individuals who we think might come in.
I haven’t asked anybody — no discussion yet — but who would I ask to be secretary of state? Who would I ask to be the attorney general? Who would I ask to be White House counsel? Who would I ask to be the chief of staff?
The one advantage I’ve had that you only can have if you’ve been a former vice president … is watching how it works, knowing what you have to have inside a White House, knowing how important it is to have a cabinet who when people look at them they radiate confidence. 
At the end of the day, the people who will join my cabinet — God-willing, if I become the president, it’s almost presumptuous talking about it a little bit — will be people who represent the spectrum of our party and who look like the country.
I really mean it — physically look like the country. Men, women, gay, straight, center, across the board. Black, white, Asian. It really matters that you look like the country, because everyone brings a slightly different perspective. In the coming weeks, we’re going to, as I said, put together a committee, looking at candidates, looking at someone to be a partner in the progress, and who is simpatico, who is someone who in the case of the vice president ready to be president at a moment’s notice.
I’ve announced and I mean it … I think it’s critically important that the vice president be a woman, with the experience and background to do the job. There are a number of women who are in that position, including women of color. The same thing goes across the board … for whomever the president chooses for his or her cabinet.
One of the great advantages I had, because I don’t think it’s an exaggeration to say most presidential historians and vice presidential historians … over the last three years have pointed out that maybe the closest president and vice president in American history have been Barack and me. Because the one thing we knew about each other to begin with, and that is that we were in total agreement on all strategic issues. We disagreed on some tactical approaches.  And the way we’d do that, we’d have lunch once a week together where everything was on the table — we met every single day — where we’d argue like hell with one another. We’d talk about what we needed to do. I’d make my case, he’d make his. He’s president of the United States and 90 percent of the time, or in that range, he prevailed. But every once in a while, he yielded. He said ‘no, I think you’re right. This is it.’
Biden noted he didn’t initially think he was the right fit for vice president and thought he could be more helpful to President Obama as a senior senator. But recalling a conversation from 2008, Biden said of Obama's reasoning: “He said, ‘it’s because I know you’re never going to walk in the White House and be intimated in the Oval Office. You’ll always tell me the truth.’
But it has to happen in private. You always have to have the president’s back. And the same thing goes for cabinet members. And there’s a lot of people out there. I can think of, if I had to, if Lord Almighty came down and said you’re president tomorrow, write down in the next 15 minutes your cabinet, I think I could do it. Because I’ve had the advantage of working with so many incredible people in business, in labor, in the academic world, in the whole public interest center, whether it’s foreign policy or domestic policy. There’s a lot of really, really qualified people who I think have the same view that I have, which is it’s not about going back to 2008 or 2012. It’s about moving ahead significantly."