Different answer depending on whether you have a product built or not. Also different if you have a consumer product. My best personal experience with customer research is for live consumer products. In that case we would recruit potential customers off of craigslist (offering $50 for an hour of their time). We would bring them into the office and sit them at the table. The entire team (small) would be in the room. We would give them a scenario and ask them to complete a task in our app in front of us. For example, you saw a really good video on Facebook made with this app and now you want to download that same app and make a video for yourself. Then we would ask them to complete that task asking us any questions they had and sharing their inner monolog out loud. Some questions we would answer but most we would ask them to figure it out. It was very important that everyone involved with building and designing the product was in the room experiencing the user test live. We would take notes on issues the customer faced and conduct about 5 interviews in an afternoon. Anything that came up 4-5 times we would attempt to fix. We would rince and repeat monthly.

Start with better understanding what exactly customer discovery is, Steve Blank and Eric Reis are good places to start.

To better understand how to conduct the discovery and what questions to ask you should read a bit of Rob Fitzpatrick's The Mom Test.

It's hard to give one piece of advice about how you should do customer discovery (without having a high quality conversation about what you are doing) b/c every venture is a different in what needs to be tested. Software ventures tend to need product customer discovery (what you should be building), while STEM companies[1] tend to need go-to-market strategy customer discovery.

[1] An example would be a company that is making an orally delivered anti-coagulant drug. They have HIGH technology constraints that aren't easily adjusted, so their customer discovery tends to revolve around questions about what their real value proposition is (is it pain free needle free drug delivery or no medical training drug delivery or something else) and other such questions.