Sorry in advance for the wall of text
Honestly, I don't know how you can all be so upset. DMC1 was a looooong time ago. Go back and play it again, then tell me with a straight face that the story was great, the characters were endearing and the dialogue was believable.
From what I've played so far, DMC5 has a better story all-around(more believable characters, far fewer plot-holes, better premise)and has fixed multiple problems with the antiquated control scheme. The combat is as tight and responsive as ever once you adapt to the new button layout. I found myself smiling at all the little inside jokes and references to the older games of the series while keeping a coherent story that actually makes sense within the context of the world.
Let's look at this in a long-view for a moment, we'll break it down one game at a time.
DMC 1

ante is introduced as a mysterious badass who accepts a job without question from a woman who stabs him in the chest, then throws a motorcycle at him before explaining anything. The dialogue started the trend of laying on the cheese thicker than a drunk roommate making nachos.
The combat was great, laying in the groundwork for the awesomeness that would come after.
The environments were precisely what was "cool" at the time, a decrepit Gothic-inspired castle where the antagonist thwarts you indirectly at every turn like a D&D Dungeon Master with a grudge.
DMC 2:Just... crap... The story was nonsense, the gameplay was lazy and the environments were a muddled brown smear.
DMC 3:A great return-to-form for Dante, and as they couldn't risk reminding anyone of DMC2, they had to go back in the story line to find something worthwhile, and decided to make a prequel about Dante discovering his Devil Trigger ability. The secondary characters were either flat or irritating, but the dialogue struck just the right balance between gravitas and cheese.
The gameplay tightened the bolts on the combat, while adding the style system to allow for many more moves and better replay value.
The environments were pretty unremarkable for the most part. It did a decent job of transitioning between the modern world and the Tower of Infinitely Contrived MacGuffins. I won't even get into the lazy design behind the chess piece enemies.
DMC 4:Now suddenly we're in a weird hybrid between modern and feudal times. The survival-horror influence of the original Resident Evil team that worked on the first one is pretty much gone, but they've done a beautiful job with the (now bright and happy) environments.
Story-wise, Dante has finished his character arc, and is pushed to the back for the sake of introducing Nero. Nero is a whining pubescent little ***** despite his ridiculous power and skill. He pines after his adopted sister, who has even less of a personality than Nero.
The combat was improved in two simultaneous directions, forcing the split and forcing the writers whip up a new protagonist who is equal to Dante in power. Dante kept an improved version of the style system from DMC3, now able to switch styles on the fly. Nero got the arm thing on the other side of the coin, and demonstrated just how much more fluid the combat could be if you could grab enemies and pull them into your combo, rather than chasing them all over the room as you bat them about.
DMC 5: The combat is a natural evolution of fusing both characters from DMC4. Dante kept the most useful moves of both characters, and added a few of his own. It's fast, fluid, and skillful, allowing for both pursuit and pull tactics when dealing with enemies.
The story attempts an update on the theme by allowing the world to exist in parallel with Dante's antics. Instead of forcing the story to take place in some far away island or tower, it's right here in our world, just on the other side of the veil. I couldn't help but smile at the thick parody of American consumerist culture, and our horribly broken banking system.
The reboot was necessary because DMC4 wrote itself into a corner with Dante passing the torch to Nero, who nobody likes.
Anyway, that's my rant/rundown. I played every Devil May Cry to date, and beat each one (yes, even 2) more times than I could possibly count or prove, and I feel that this is the best possible direction in the series. You are all, of course, encouraged to form your own opinions, I only ask that you make them as educated opinions as possible. You don't have to agree with my views, but please take the time to consider what other roads were open to the dev/writing teams before you scream about them taking a wrong turn. We might have ended up with a new trilogy about Sparda's grandchildren, or an entire series surrounding Nero. In the end, the writers painted themselves into a corner, and laying down new floors left the fewest footprints.