My favorite run of Inhumans was the War of Kings stuff. I might have read it due to the heavy X-Men ties. If not... I recommend the following read order:
If you haven't read them, I recommend reading the following in this order to get an understanding of what's come from the X-Men side of the War of Kings story. The Inhumans don't come into these, but their story and this story collide in War of Kings.
X-Men: Deadly Genesis 1-6
Uncanny X-Men 466-471 (The End of Greys)
Uncanny X-Men 475-486 (Rise and Fall of the Shi'ar Empire)
Emperor Vulcan 1-5
If you want some lead-in with the Inhumans, I recommend the following. All are good, so if you're down with really giving them a go. Give it a shot:
New Avengers: Illuminati (one-shot)
Son of M 1-6
Silent War 1-6 (there's a jarring transition from the end of this to WWH, but it's explained later)
World War Hulk 1-5
Then the two above sections mold into what follows:
New Avengers: Illuminati 1-5 (or at least 1 and 5... you can skip 2-4)
Secret Invasion: Inhumans 1-4
Secret Invasion: War of Kings (one-shot)
X-Men: Kingbreaker 1-4 (No Inhumans here, but important toward what came before in the X-Men comics I mentioned)
War of Kings 1-6
War of Kings: Who Will Rule? (one-shot)
There are a lot of tie-ins to the War of Kings, so if you like what you read above, you can branch out and read more of what's going on with other cosmic characters like Nova, Darkhawk, and the Guardians of the Galaxy. But focusing exclusively on the Inhumans, what's above is what's important.
And if you're wanting some follow up to all that's come before, read:
Mighty Avengers 27-31 (This wraps up some stuff from Son of M and is referenced in the Realm of Kings mini that follows this)
Realm of Kings: Inhumans 1-5 (This shows what they're doing in the aftermath of War of Kings.)
Then from there I hear they're scattered through Hickman's Fantastic Four run before Blackbolt becomes prominent in Hickman's New Avengers (the first 6 or so issues) and the Infinity event crossover. I've not read Hickman's Fantastic Four, but Infinity is absolutely stunning.
*edit: I think this is pretty much the core for of what happens to the Inhumans between the 2003 series and the modern stuff with the Terrigenis cloud.