Always remember your boss is not your friend. That doesn't mean they are your enemy, just not your friend. Your relationship, like most relationships in business, is partly cooperative and partly adversarial. During everyday work it is easy to focus on the cooperative bits and forget about the adversarial ones, but pay and conditions are intrinsically adversarial.
The conventions for navigating this mix of cooperative and adversarial stuff are known as "professional behaviour":
You owe your boss (and by extension your employer) your best efforts during working hours. Full stop.
Your "true feelings" are irrelevant to the business relationship you have with your boss. Leave them at the door.
Having an employee leave is a routine hassle for your boss. They won't welcome the news, but it's their job to deal with it. It is not your problem.
The professional way to deal with this is to provide your boss with maximum warning of your departure consistent with safeguarding your own interests. Therefore you should get a signed contract with your new employer first. Until then you are still negotiating.
Once you have the signed contract, write a short letter to your boss, CC the HR department. State that you will be leaving the company and give the date you plan to leave (obviously, after your notice period expires). Thank them for the improved offer but say that you have decided to leave because you feel that your new post is better in line with your longer-term career aspirations. Say you have enjoyed working there and wish them well for the future.
(The last bit is standard fluff/politeness; part of those "professional behaviour" conventions I mentioned).
Ask the boss for a meeting. Tell them the key points and hand the letter over. Keep it short and impersonal. If your boss is behaving professionally they will accept this, say how sorry they are to lose you, wish you well, and end the meeting.
If the boss starts to probe for reasons or suggesting a pay rise, shut the conversation down by making it clear that your departure is non-negotiable. You have already signed a contract with your new employer, so increased offers don't matter. If the boss asks personal questions about happiness or stuff, stick to the line about your career ambitions. Keep it generic, even if that sounds awkward. Remember that if your boss really valued you that much they would have made an improved offer before it got to this point. If it really gets sticky then end the meeting yourself; "I'm sorry, I don't think this is getting us anywhere, so if you don't mind I'd like to end this."
After the meeting, send a copy of your resignation letter to HR.