It isn't easy, and those making light of it here have no idea how challenging it is for a manager, even if it is the right thing to do.
In some cases you're making someone unemployed, knowing they can't pay their bills, and knowing they will struggle to get work. That is a difficult thing to do.
The first thing, is making sure you have followed to the letter, the HR process in your organisation and that you have documented everything.
By everything I mean dates, times, words and phrases used, copies of signed written warnings, the lot.
Any conversations with the individual will have been documented and shared with them already I presume, so you have a written basis to show the situation already if you need to in an employment tribunal.
Depending on whether they have worked for the organisation more than 2 years is the next issue. If not, its 'easier' to do, they have very little rights other than if they can prove they have been discriminated against. If they have, it becomes a bigger beast, and you want your HR team to be dealing with it, or at least taking the lead.
I recently had to advise someone they wouldn't have a job anymore, and I actually called ACAS in the weeks leading up to the inevitable to get their view. They said everything done had been done to the letter of employment law, and at the very least was 'legal and fair'. You can't really hope for much more than that.
When it comes to telling them, less is more. A simple 'as you know, we've been trying to support you through performance improvement, and we've decided unfortunately that you aren't making the progress we had hoped. As such, we will be terminating your employment immediately', and just hope you get a 'reasonable' reaction. Don't get drawn into giving lots of examples. At this point, it is too late, and this isn't a discussion, this is a statement.
Reasonable reactions are probably still tears and swearing, but its not all bad. The pressure it removes you from is incredible once they are escorted off site.
For goodness sake though, make sure you do the sacking with someone as a witness, and share a write up of the event with each other with the other person replying 'I agree this is a fair and honest reflection of events' so it is there as evidence.
The important thing to just remember, everyone is human, so do it decently and with respect. You never, ever know when you will meet someone again.
Do not go parading around the office like a god afterwards. Simply doing the act shows people you'll take on difficult decisions. Your team will respect you if it was the right thing to do.