How would you change a product team/organization to be more KPI driven?

What a terrific question! This is one takes time and depending upon your organization will require patience.

The best way I have learned, is to ask "why" a particular feature is being prioritized, and the "impact" it is bringing to the customer. The best way to create change is to start with your own product team (even if that consists of you as the PM, engineering, and UX/UI). For every epic/story, make sure to note down the KPI that it will impact, and why that is important to your target (end) customer. 

After a few cycles (e.g. sprints if you follow the agile approach), share the results of how your product work (launched features) impacted the KPI you were measuring, and why that is critical to your overall business.

Evangelize this within your org, and see if there are one or two other teams who are interested in following suite. 

The more you share and report broadly, the easier it is to create change internally.

Becoming more KPI driven is a matter of desire and taste. No person, team, or organization attempts to change without believing that behaving differently will result in an improved outcome they care about. It's only possible when leaders buy into how it would improve the success of their teams and business (e.g. profitability, valuation growth, employee engagement, talent retention, positive social impact, etc.) Some companies are steadfast that the use of KPIs should not equate to being data driven everywhere in the company. They prefer to have data informed teams that reserve room for intuition and qualitative insights. There is no right answer here.

If we find ourselves with a company that's bought into a shift towards being KPI driven, but is trying to figure out how at the team, group, or division levels, then I'd recommend the following: 

1. Have the leaders of the team/group/division define their strategy for a period of time through written outcomes, assumptions, and principles that are most critical to their success.
2. Gather all the data already available and audit it for quality and trustworthiness, then see if you can model your product or business (i.e. in a spreadsheet) to see if the assumptions you've made and outcomes you've articulated can be explained by your data. If not, note what's missing and how you could gather it (and be comprehensive at this stage).
3. Work with your engineering and/or data team to instrument the metrics you need, backfilling where possible. Remember that you'll need continuous energy to ensure your data remains audited and accurate, as data corruption can severely disrupt your KPI-driven organization.
4. Develop a process for regularly collecting, analyzing, and reporting on the chosen KPIs. Without this ritual, your efforts will be for not. Being KPI-driven means knowing and using the data to make decisions. In my experience, to get the flywheel spinning, you need to have weekly rituals that can morph to monthly rituals. These can be augmented with quarterly business reviews.
5. Make sure that the chosen KPIs are easily accessible and understandable to all members of the teams. This may involve creating dashboards or other visualizations to help team members quickly see how the product or organization is performing. Repeat your KPIs at kick-offs, all-hands, town halls, business reviews, and anywhere else you gather. It's only when you think you're over communicating them that you've probably approached a baseline level of understanding of the KPIs, and how they inform decision making, across your company.
6. Provide regular training and support to team members to help them understand the importance of the chosen KPIs and how to use them effectively to improve the org. If you have a wiki, put your tutorials there. Make it mandatory to consume these during onboarding. Offer self-serve tooling. The more people can be involved with the data, the more you'll make this cultural shift.
7. Regularly review and adjust the chosen KPIs to ensure that they are still relevant and useful. Account for any changes in your outcomes, assumptions, and principles. Assess suitability annually. Set targets annually and adjust mid-year. Some companies do this more often, and your culture should dictate what's best.
8. Lastly, make sure that all KPIs have their lower level metrics clearly mapped for the company to see. Teams influence these input metrics more quickly, and the mapping brings clarity to decision making.

To make a product team/organization more KPI driven, it is always best to start at the top. The Head of Product at your organization should provide the broader PM organization with the strategic context of what business objectives are essential for your team to hit. 

From there the PM team can take that business objective and break it down into their own objective and key result.

If you are a product leader seeking to make your organization more KPI driven, what I would do is identify what metrics matter to your business and put those into individual PMs goals. For instance, let's say it's critical for organization to ship Product Y at the end of Q2. PMs Bob and Anna are responsible for Product Y. Then I would make it part of Bob and Anna's individual performance goals that they deliver Product Y by the end of Q2. Of course, there are going to be many reasons why Bob and Anna may not be able to deliver Product Y (the big one being engineering or design). And while these reasons may be true by clearly stating in their performance goals that they are accountable for that deliver this ties Bob and Anna to ensuring that outcome occurs and that they keep you posted on what needs to be done to reach that outcome. 

If you are an individual contributor seeking to make your organization more KPI driven, I would start with your own product. Ensure that you are aware of the key metrics your organization values (e.g. Annual Recurring Revenue, Daily Active Users, etc.) and determine what key results your product needs to hit to help your team reach that outcome. I'd then focus on sharing your KPIs with your manager and peers in engineering, user experience, and product marketing for alignment. With KPIs aligned, I'd regularly update others on your metrics and show progress. Others will notice and likely start requesting that your peers follow.