I've been looking at this both for clients and personally so have some relevant experience.
In my view, the biggest challenge, and most important aspect, is right at the start, ie to determine what "problem", "pain" or "need" your product or service is helping the customer address. This needs to be very specific and not expressed in general terms such as "improving poor customer service" or "reducing product returns" or "increasing revenue" etc. You really need to explore the details of how the "pain" is actually felt and how your product/ service helps. Put yourself in the position of the customer. The more specific the better because subtle differences here can lead to major differences as you go through the process and lead you to a different target which may be way off the mark. Especially as this is a multi-stage process, the early steps have a major impact on the eventual outcome.
The goal is to ensure that when you get to your target market they really resonate with your solution and it's an easy decision to buy or at least investigate further.
Happy to take a call or to handle follow up questions.
The most important part of "customer discovery" is really understanding what keeps your ideal customer up at night. What is his or her's fears, goals, and interest. In other words creating a mind map of the "Ideal Customer". When you create a mind map you will then discover a better way to market to them. You will have the ability to become part of their daily lives and not just advertise at them but become a trusted resource and an authority. This will allow you to have them coming to you asking for help or purchasing from you. Your gross profits will rise and your advertising cost will decrease. To put this in perspective... really understand your ideal customer and you will be able to start talking their language vs trying to sell them on a lower price than the competition. Now this is only one type of customer discovery but you would need to clarify more. There is also customer discovery once you have them wanting to buy which that is called a BANT discovery e.g. Do they have BUDGET, Do they have the AUTHORITY to purchase, What is the NEED vs a want, What is the TIMELINE. Some say BANT is old school but often people want to change for the sake of change. However people purchase for the same needs and wants as they did 25 years ago. The only thing that has changed is how we find it and purchase it. Hope that helps and if you want a Mind Map guide just search for me and I have a free download help guide on my site about Mind Maps and Archtypes.