Like I said, how you integrate it in your story ultimately depends on how you want to spin the entire concept itself.
For example, one of the concepts I was spinning in my head for a while was to initially start off with a sort of entry arc to sort of ease into the concept. The intial threat is not at first the actual beings from hell, but those connected to them. Possibly another variant on empowered humans, mayhaps seeking other empowered humans to in some way boost their own potential or some other means. Either going into a Demon Possession to gain power sort of deal or a Demon Summoner route.
The later actually appealed more, though you could possibly spin both in a similar direction depending on how you set up the rules for how the powers worked. For the Demon Summoner route, perhaps set it up such that the more they start asking of their summons, the more they must 'give back' in return. Which naturally leads them to needing to track down other empowered humans to give them in order to keep their contracts without losing their own lives in the process. You could spin demon possessed in a similar fashion (hell, it's even possible to blend the two concepts together and just have it be a more potent version of demon summoning).
Use that as the initial framework the spark the first confrontations, and then put the big wrench in things to really set an actual 'Invading/Escaping Hell' plot rolling. Multiple ways to go about this too in fact. The first, and more contrived choice admittedly is to have some personal element to it. Maybe you have one of the summoned demons recognize one of the group in some way. Spin it off Chad's Right arm of the Devil, maybe go into the Chad's Grandfather plot that I've been dying for.
Another potential route that I had was a demon recognized Zangetsu from a long past fight thanks to the reincarnation concept. Which lead to his interest in Ichigo, constantly breaking out personally to try and entice a fight until he finally has enough and goes full in to stop this guy once and for all before someone else gets hurt as he tries to force him to fight.
Lately though probably the most interesting route is actually the Demon Mark concept. Similar to what I've heard Berserk uses in some fashion (Though admittedly I'm not terribly familiar with that series so I may be confusing something horribly), perhaps in the initial arc with Demon Summoners, though the group wins and manages to drive them off the demon summoners do succeed in phase one of their plan and engraves the members of the group with a demonic mark somewhere on their body. Not a curse specifically to the user in and of it self, it doesn't really affect their powers or has them become part demonic now*.
No instead the function if this mark is that allows demonic beings to continually break through from hell into their general area to attack them, basically making them living bait for hellspawn and in turn constantly put everyone around them in constant danger. Were that not bad enough, the only means for removing this mark exists deep in hell itself, where the vast majority of the creatures that desperately want to eat them now reside.
*Edit: Though this being said, you can again spin this multiple ways. Maybe Ichigo is the only one marked, and the rules are a bit different and instead it significantly weakens the one engraved and in turn makes it supremely difficult for them to use their powers effectively. Meaning he's now desperate and really needs the rest of the group to pull their own weight or they'll get destroyed. Maybe only those who have been marked can pass through the barrier to hell at all, meaning they can't get shinigami support here. Hell maybe the mark slowly drags the person into hell and the only way to stop it is to kill the demon who engraved it before the person eventually becomes a fallen soul, unable to ever escape from hell.
Lots of ways to spin it to service the plot, and in turn a lot of fun ways to develop the rather simple idea of a Demonic Mark being the plot device to have them go to hell, rather than another rescue arc or random adventure for the lulz.