REASONS TO DO IT
It's almost essential to get as far as you can in the Crusade each day; nothing else generates as much money, and in the mid-to-late game, you will always need more money to skill up your characters. It provides other loot drops, and sometimes you luck out with them, but money is the main thing.
Crusade coins drop with the purple and final (gold) chest; 100 for the first, 200 for the second, 300 for the intermediate ones, and 400 for the final. They can be redeemed for soul tickets for Crusade characters (some of which are very tasty), as well as some other chaff, but soul tickets are more likely the target.
You progress through 15 battles, offering progressively higher rewards with each. Any damage your characters take carry over to future battles; so does any energy they possess at the end of a battle. A character killed in a battle is killed for the duration of the Crusade.
Corollary: Characters who regenerate (like Phoenix and War Chief) only regenerate across the crusade, not from battle to battle. So a War Chief who has died and regenerated can die only once more before permadeath. A good idea to heal him, if you can, before he dies the first time.
Even if characters have died in a battle, you can RETREAT before the battle ends, and those characters are not dead yet. So if things are going badly, do so, and try with a different team.
SELECTING A TEAM
Your team should contain Chaplain. ALWAYS. (Possible exception: Shadow Shaman also heals, and can replace her, but should generally be used as an emergency replacement should you lose your Chaplain).
Because damage is retained, Chaplain is one of the few ways of healing your characters back up to functionality. Only very rarely use Autofight, and be strategic about when you heal. You also need a good solid tank; Brute, War Chief, and Death Knight are all excellent choices. In fact, you probably cannot complete Crusade without a Death Knight – but this can be one you hire from your guild, rather than one you own. Also, you should try to get through as many battles as possible (at least 6 and hopefully as many as 9) with other tanks, to save Death Knight for the more challenging levels.
(Death Knight and Death Mage drain health from other characters, and survive a bit better in Crusade; the in-game tip says you should use one or the other. However, while Death Mage is a very cool DPS character, Death Knight is key.)
Two slots should be reserved for your best DPS characters, whatever they may be.
The final slot can be either another awesome DPS character; a crowd control character; or a character like Mystic, who lends energy to other characters (and does reasonable damage himself).
TIMING THE CRUSADE
The power of the enemies you fight in the Crusade is determined AT THE INSTANT YOU TRIGGER THE RESET. Thus, in the morning, the first thing you should do is reset the Crusade. AFTER that, skill up your characters, do all the Daily things to gain Team XP, spend your stamina, enchant something, equip things – and, ideally, gain a team level and level up your main team.
This way, your team will be a bit more powerful than expected by the Crusade's algorithms, and will have a better shot at success.
The first few battles should be easy for you to win. Consequently, don't use your A team; use characters from your B team, reasonably high level and skilled but not your best. Try to avoid triggering any skills; bank the energy. (However, DO trigger Chaplain's heal toward the end of the battle, if any characters have taken substantial damage, to heal them back up.)
At the end of the battle, swap these characters – who, ideally, now have max energy and still max HP – for another set of characters. The characters you energized can now be used later in the Crusade, already set up to trigger their skill, when you get into trouble.
By the end of the third battle, you most likely have all your useful characters maxed out on energy. (And it's okay if you lose a wimpy tank or something in the course of setting this up.)
At this point, you should be fielding your A team. At the start of a battle, trigger two skills – typically those of your two middle-rank DPS characters. (If the rear character is also DPS, they tend to regenerate energy more slowly, so probably best to keep that for an emergency, at this point.) It's probably okay also to trigger the skill of your tank.
Your goal at these levels is to kill expeditiously; heal up so you remain at or close to full health for the whole team; but never to have a character use their ultimate skill twice in the same battle. You want them to start the next battle with a full (or close to full) energy bar.
A note on healing; healing occurs over time. If you trigger heal just before the last enemy dies, you'll only get part of the full benefit from healing. It's better to heal a little earlier in the battle, and get the full benefit – or even to wait to the next battle, and heal at its start. This last strategy is dangerous if facing Phoenix, however; in that case, you want to wait until Phoenix fires, and then heal, because everyone on the team is going to take damage.
These are the levels where it starts to get hairy. Still, for as long as possible, stick by the rules from battles 4-6; trigger two DPS skills – and now your tank's skill – ASAP; heal as necessary; use the left-most character's skill only when you think it necessary. Try to end with energy bars full, but if a battle is proving tough, absolutely, trigger that skill a second time.
Try to avoid losing characters, but if a character ends a level low, Chaplain also ends low on energy – and, particularly, if you're facing a team with Phoenix in the next battle, so you want to save a heal until Phoenix fires – swap out that team member for someone from the B team with good health and a maxed energy bar.
When facing a team without Phoenix, consider swapping them back in and doing an early heal.
Some point in here, it is likely that your best tank will look a little frail, or even be dead; that's when you hire a Death Knight from your guild, if you possibly can. Or another decent tank. You only get one hire for the duration of the Crusade, so be careful about this.
If you've gotten this far, you've done well. Take a breather. Step out of Crusade, and maybe skill up your remaining characters a bit. Do a Trial and collect that last Daily event, perhaps. You've just earned 900 Crusade stones, so this is a pretty good day even if you get no farther.
But it's still worth going on, because the monetary rewards for the last three chests are nice, the final one has another 400 Crusade stones, and the other loot drops are on more valuable random tables – no guarantees that you're going to get something cool, but a slightly higher probability than with earlier battles.
One thing to remember with these battles is that enemies ALSO retain damage. So if your team gets totally wiped, but does reasonable damage to the enemy, you can come back in with your B team and maybe win the battle that way. Or a third time around with your last-gasp C team. And what do you have to lose, really? Give it the old college try.
At this point, trigger everything you can the instant the battle starts – well, usually. Keep Chaplain's heal for when you need it. And if facing Phoenix, keeping an attack that can disrupt it mid-beam after they fire can be valuable. You are now facing teams that may be more powerful than yours, but start with no energy; your edge is getting off your skills early, or at the right instant.
CRUSADE FOR NEW PLAYERS
When Crusade first unlocks, you will not have a B team, let alone a C one. Realize that you're not going to complete it, unless a miracle happens. But you are getting money and Crusade stones as you go, and that's not bad. Get as far as you can, and don't sweat it. You'll get there, eventually.
CHAPLAIN, CHAPLAIN, CHAPLAIN
She is (along with Death Knight) the essential character, and you have her, since she's one of the starting five.
KEEP HER FIRST SKILL AT MAX. That's right, forget about evenly distributing skill points across characters or something; always keep her heal skill at her level (and keep her level at your team level.) Give yourself every edge in Crusade.
Prioritize her for enchantments. In general, don't spend resources enchanting, but do the Daily event that asks you to enchant, to get the reward; enchant up Chaplain's gear before anyone else's.
And, oh yes, grind for her soul stones, every day, and star her up. And promote her whenever you can, and grind for the necessary gear.
THE ARENA MYTH
I have seen promulgated elsewhere the idea that the toughness of the enemies you face in the Crusade is based on your Arena ranking (with the idea that, by nerfing your Arena rank, you can face wimpier opponents in the Crusade). I have tested this proposition, by purposefully lying low in the Arena, then zapping up as high as possible in a short period of time, and seeing whether Crusade changes. It does not. Do as well as you can in Arena to maximize your Arena stones, and don't worry about this.
There is a relationship between Arena and Crusade; the teams you face in Crusade are the teams players have selected for the Arena. This makes sense, and is an easy way for uCool to select teams. But the teams you face are based on your power, not on your arena ranking.
GAMING POWER FOR CRUSADE SUCCESS?
I hypothesize, but have no real evidence, that it is possible to (modestly) improve your Crusade success by gaming the power system.
A character's power is based on their stats, which is a function of character level, stars, promotion level, skills, and gear.
The one thing on that list that you might be able to game is: Skills. If you look at a character, and think a particular skill is not likely to help you in the Crusade, you could avoid spending skill points on it, keeping it at a low level. This would make your character look less powerful to the game system – but potentially as useful to you in Crusade.
My guess is that this is the only potential way to play with the odds, however.
(Maybe you could also craft but not equip a particular item of gear, if it increases stats you consider non-essential, But keep it around so when you have all gear slots fillable, you can promote.)