Most of your battles should go something like this (on 3x, 6x speed):
Integrity
Huns get +1 to integrity from every war they fight. So, declare war on everyone you see that you don't care about being friendly with. Ask WRE to join their wars (for a fee, of course) or anyone else for that matter. Do this starting form turn 1. Currently, I am playing a "this is total war" legendary Hun campaign and I am at war with 30+ nations at all times. My integrity is going up by 30+/turn just form those wars.
Money
Playing a conventional campaign, you can get quite a bit of cash from subjugated nations especially if you help them expand. Playing "this is total war" style does not have this income option.
The other tip I already mentioned: get to Mediolanum asap and raid the trade route there until either someone destroys the city or your treasury is around 400K large.
Third tip: just build econ structures in your hordes: Yurt builder is great as it gives cash + growth; sheep farms give food + cash; etc.
Fourth tip: don't upgrade your horse archers (avoid researching the tech that does just that). Tier 1 horse archers are all you need. The later ones get slower anyway and having tier 1 horse archers as the bulk of your army saves a ton in upkeep.
And another one: do not raise your taxes to very high at the start of the campaign. That will stifle your horde growth. At the beginning try to have more or less balanced budget from turn to turn and build from the money you get from sacking. Raise taxes only in emergencies or when you are confident you do not need extra growth. By that time though, you'll be rolling in cash even on normal taxes.
Campaign strategy
Don't be afraid of those multi-stacks following you. If they get too close: attack with your nightfighter generals and destroy them one by one. Otherwise, if they attack: always defend and only retreat from the battlefield if you're out of ammo before the enemy breaks. Other than that: sack and raze everything you do not subjugate. Avoid winter-areas until you get the tech giving you snow immunity.