What's the best way to break up with someone respectfully?

Partial rather than full honesty is likely to lead to a more respectful conversation You can't kindly say that you can no longer stand his/her personal habits or that you have found someone who is much more attractive.

What you can do, respectfully, is list the things you've enjoyed about him/her and thank for any nice actions you've benefited from. there should be many, or you would not have been together for more than a week or two. You then say that you don't see the relationship going on and upwards.

You have realised that it is not fully working for you and you would like to part on a friendly basis, but you don't expect to go out with him/her anymore. Return his/her possessions properly and express regret but say there is no blame intended. You have probably left him/her short to the same extent. Ask that there be no bad feelings, only shared regret.

The best way to break up with someone respectfully is to have a face to face conversation with the person. Don’t do it publicly, find a place where you both can express your emotions freely.

Be honest about why this is happening, answer their questions, and do not leave questions hanging in the air. Be clear about what you gained (the positive) from the relationship, in as much as you want to dwell on the negative, speak to them with warmth and love for what you had and be clear that your relationship is over.

Do not break up via text messages, this comes across as cold and uncaring, it doesn’t give room for the other person a chance for a proper closure.
Hope this helps.