The first four mean the same thing, but
более красивый - I would never use. Sounds grammatically incorrect, or rather "как-то не по-русски". Although, it is ok to say: менее красивый, чем
красивее - your number one choice, use this one, and you will never get it wrong
красивей - good enough for informal conversations, especially if you want to sound like an ordinary Russian
краше - could be used, but usually as a matter of figurative speech or something like that. If not sure, do not use this one or you may end up sounding weird.
Safe ways to use it:
1) Let's say two children look messy as if they played in mad or have chocolate all over them. You can say with sarcasm: Один краше другого or одна краше другой
2) Proverb: краше в гроб кладут. Be careful, it sounds very offensive. May be used when someone looks very sick and ugly. Usually said behind someone's back or a mother may say it to a daughter if trying to make her take better care of herself.
3) Pay a complement when flirting with a young girl: краше тебя нет никого, but say it with a smile; mean it, but don't be too serious.
The following two mean very much the same; the difference is hardly noticeable:
самый красивый -- use in every day life or to point out the best looking person/item, but say it as a matter of fact, with no emotion. Also, use when pointing out / selecting the best one in a specific group
1) Вот тебе самое красивое яблоко (when selecting out of a box);
2) Самый красивый цветок -- у Наташи (assuming there are more people and more flowers);
3) Лондон -- самый красивый город из всех, где я бывала
красивейший -- ok to use with passion and when making a general statement, but avoid using when simply comparing things.
1) If one said: Лондон -- красивейший город, it would mean that one finds this city beautiful or one fell in love with London. It does not mean that one is actually comparing it to the rest of the world
2) Наташа нашла красивейший цветок! Means that she found an unusually beautiful or rare flower, again, this is not so much to compare, but merely an expression of the feeling.
3) If you took an apple out of a box and said seriously: Вот красивейшее яблоко, it would be like saying: "how do you do?" when it is more appropriate to say: "hey, what's up?"