Letting someone go, regardless of the reason, is never an easy task. You are making a decision that is going to have an impact on someone's life, maybe even affect their family. However, when you are responsible for a business, you have a responsibility to make decisions that are in the best interest of that business. Would you rather have the business fail? That wouldn't help anyone.
I've had to recommend letting someone go. I gave her every opportunity to prove herself. I was never unkind about it and felt it important to be honest with her. So many poor managers, colleagues and even personal friends often find it difficult to be honest when dealing with another person's weaknesses to their face which leaves that person upset and wondering why they were treated that way. You would probably too if you were in their shoes.
About 5 years later, out of the blue, she stopped me on the street one day. I didn't remember her at first. We had only worked together for a short period of time, maybe a week. Then she proceeded to thank me for changing her life for the better and told me that my recommendation pushed her to go back to school for a year, learn the skills she really needed and that she was now successful, making more money and much happier with herself and her life. She had been hoping to bump into me again one day to thank me.
I've had a few other similar experiences over the course of my career. Once someone's mother even call me to express gratitude for changing the course of her son's life in a significant way -- a year after he stopped working for me. Again, this was out of the blue -- I had never met or spoke with her before. In this particular situation, I had managed to avoid reaching the point of having to fire him.
I guess I am very fortunate that these people always did something positive with the feedback I offered them. Some people will never take it well. They will blame you or anyone they can find but will never take responsibility. I am not sure that there is anything you can do to help that type of person because they don't want help, just opportunities to blame others.
The way things work out has a lot to do with the kind of manager you are and the relationship you've developed with them.
If the person was good but just not good enough for your specific requirements, it might help you feel better if you do something to help them find another job where they would be better qualified, or show them how to get the training they need. Remember though that you should limit yourself to helping them. Don't try to do anything for them. Their best can only emerge from within themselves. Help this person only if they want to help themselves.
Just be sure you've talked with them about their situation and try to get agreement on the best course of action. It can help if you can get them to see how they might not be a good fit for the role and your business requirements. Help them see things from your perspective and maybe ask them how they would handle it if they were in your shoes. Remind them that you would like to maintain a good relationship with them and potentially work with them again in the future when they have more experience. That should help prevent things from turning ugly. If they do get angry, you can rest assured that you did everything you could for this person.
Do your best and treat them with the dignity and respect you would hope someone would treat you. In the end, that is all anyone, including you and your employee, can do.
Hope something in all of this helps. Keep what makes sense to you in all this and forget the rest. Not everyone is ready for all of this advice.