Basically, you need to complete 5 Stories for any character.

Pretty much the same, as the previous.
But now to complete all stories, except for Lost Warrior (he doesn't have one, it seems)

You need to buy literally everything in the Gallery
All art, music, movies, voices, you get the idea

There are over 100 Illustrations, which costs 100 gold each. Then, Character Voices cost 1,000 and there is one collectible per character. For the BGMs part, you'll need to buy 15 tracks that cost 500 each. Lastly, there are only three movies, but they cost 3,000 gold each — plus, you need to complete the Story mode to unlock two of them.
Personally, I'd recommend using some software that could spawn you ton of money.

300 Online Matches, it does not matter, win/lose, ranked/room.
If you like playing with people you will get if eventually.
If not - WELCOME TO THE JUNGLE.

Defeat the strengthened Lost Warrior in Arcade mode.

So basically, its an Extra Unlimited version of the Lost Warrior.
Just dont die/lose a single round and Good Luck with that one.

Complete 10 matches as Lost Warrior (you need to unlock him first, ofc)
And thats basically it.

I do not see any reason in explaining any other achievment, since they are mostly self-explanatory.
Record some moves in training mode, play there for an hour, complete all combos and challenges for one character (I personally recommend Crusader for this one). Grind a lot of Online, play arcades, story mode, try out every character and so on.

Enjoy the game, folks, have a nice day.