Here are some thoughts based on my experience. Generally speaking, this person is an adult and should be able to handle direct, clear messaging about whatever issues exist. It may make sense for them to be surprised when the final action comes, but they should not be hearing the reason for the first or even second time. They may ask at the time of dismissal for you to reconsider. I would not recommend doing so. Once the decision is made, follow through.

The details may all vary depending on the size/maturity of the company, your location, and internal attitudes toward such dismissals. Partner with HR. The company may have specific processes they want you to follow and/or there may be laws in your country/region that require specific actions for compliance purposes. They also can help you understand the prevailing attitudes toward these situations among company leadership.

When the time is right, make sure IT is in the loop, or whoever is responsible for controlling employees' access to systems. You'll also want to have a plan for collecting their computer and any other company-owned equipment and materials. Things will vary based on the above but also based on the core reason.

Cultural fit is probably not something the person can be coached into easily, and when that is the core issue, it is common that relationships have already eroded to the point of extreme strain. In that case, a direct conversation citing this with a termination date may be the right choice. It doesn't have to be immediate dismissal, assuming you don't expect that person to go on a "scorched earth" campaign. This is really a situational call but the safest bet would be to just make that conversation be their last at the company. Again, they may be surprised with the action but they should not be surprised with the reason/issues.

For performance issues that in theory could be corrected, I would suggest being very deliberate in communicating the specific areas of concern and expectations before taking any formal action like a performance plan. Ideally the goal of such feedback is to be constructive and in the spirit of helping the person succeed. Sometimes improvement plans are used as a box-checking exercise, I'm not a huge fan of that but it can make sense in certain situations.

If you have concerns that the person may react negatively and things may devolve into a verbal or physical altercation, you'll want to think carefully about who to include in the final discussion and plan accordingly to de-escalate. If it makes sense, help them with their transition. This could mean allowing them to continue working for a few extra weeks, or helping them find another role. I wouldn't suggest trying to get them transferred to another department unless it is very clear they are a good cultural fit and are well suited to a different function that needs help.

This process is obviously tough on the person but it's not easy as a manager either so seek out support for yourself. If you are confident it is the right thing for the team, it is your responsibility to act, but try to do so with grace. Good luck.