All in my opinion, as the thread was made to showcase:
#1. Rewrite the original series, I do NOT want a DMC vs DmC debate, but imo it had a lot of redeeming qualities, but I was not a fan of the very different backstory and mannerisms for Dante and when compared to DMC, it felt way too much like Supernatural's low-budget depiction of mobsters (demons, etc) which does have its appeal, but I have a love-hate relationship with DmC.
#2. Make it keep the maiiiiin characters that have solidified their presence in the series. (Eva, Sparda, Mundus, Dante, Vergil, Enzo, Lady, Trish, Nero is debatable)
#3. Make it a long-ish game that feels very Devil May Cry with enough introductory material to cover Devil May Cry 3's role of retroactive character development. Make it feel like the first few volumes of a comic book; Let me jump right into slaying demons, but pop enough cutscenes and/or flashbacks with gameplay that Sparda's story is solidified and I feel emotionally invested in Dante's motives for becoming a demon hunter.
#4. Keep most memorable details like Alastor (would be nice to get audible dialogue for both of them, I very much liked that all of the Devil Arms in DMC3 having characterized baggage, would be a hilarious nod to VJ if Alastor were voiced by his VA in VJ) also Ebony & Ivory and Tony Redgrave, reuse the history in a believable way that fits canon as needed.
#5. Go with the same flow gameplay has had up until DmC. 1, 3, & 4 had awesome gameplay and 2 had some fun gimmicks (mostly devil hearts) so don't change much there.
#6. Try to let us still believe its the same characters, only deeper. Hence, keep whatever voice actors are possible.
#7. A whole bunch of what everyone else said. Keep the vintage action-movie vibes, keep the 'quasi-horror' themes, keep Dante's image roughly the same, but imo he should basically be like DMC4 Dante, basically a serious but wise-cracking demon killing Spider-Man like character. Give Dante (and others) personal and physical development. Just like the old games made you jump through hoops just to get A that allowed you to go through Door B to go to Place C and get Item D (you get the point) Dante & co should actually struggle with becoming stronger characters. Throw more curve balls. If the enemies are hard for players, I want Dante (or other) to sympathize and be winded like I am. Also give said enemies good reason to be that hard more often. The only success of an enemy getting the jump on Dante who isn't A. Mundus or B. Is either vaguely or completely related to Dante is Bolverk (and debatably Jester, as he was able to take care of Dante using outside forces to compensate for not actually being in Dante's league)
I'm sure there's a whole lot more I could add and elaborate on, but that's as far as I can go for one night. It all really boils down to HD remaking the first DMC but incorporating familiar plotlines from 3, 4, and 2 in a way that feels complementary to the first installment.