How do you fire an employee in your startup ?

Well your question is twofold: On one hand, your legal standing and responsibility as an employer with respect to whatever jurisdiction you're working in. The second is the more personal relationship an entrepreneur has with a member of the team.

The first one might be easier to answer - a chat with your lawyer should tell you if you're safe letting the employee go, and what liabilities you incur in doing so. Example: In Jordan, an employee can file for wrongful dismissal if not warned three times in writing, and is hence entitled to end of service compensation. You also need to go back to your employment contract and go over the termination clauses to check for any specifics that were outlined there.

I'd also recommend having a policy in place before you do this. Example, what is your policy on whether that person continues to work past your 'firing chat'? In the case of, say, a developer, still being around sensitive code is a liability you want to avoid. A clean cut is always easier. What is your policy on compensation?

On the personal level, it is never an easy task. At a start-up level where you're doing the hiring and firing (as opposed to a larger operation with a dedicated HR officer), it almost feels like firing a family member. First and foremost, be 1000% sure of your decision. This will surely have ripples among other members of the team and how you handle this will impact employee perception.

Objectively speaking, I'd say you have to be kind, but firm. Explain why you feel that person is not a fit in your company, and yet stand your ground and stick with your decision in case of pleas and promises to improve.

If you really like the person but just don't see a fit within your company, you can always offer to provide recommendation letters and perhaps even make an introduction or two to companies you feel would match the qualification/skill-set.