First thing, let's talk about this special enemy status. You might notice some enemies start to glow yellow/white and steam raises from their bodies. When that happens that specific enemy is going berserk and he can't be staggered by attacks except for fully charged Eryx attacks or a parry. That means you can't stop him from attacking unless you let your Eryx gloves charge twice or strike at the same time as he does. Some enemies like Ravagers can be stunned while going berserk by Aquila's Round Trip but not many others.

That being said remember that heavier enemies like the Butcher can't be staggered by normal attacks even outside of berserk mode.

Use any of the many collectible guides before attempting SSS ranking a mission. Do it before the SSS run because you will surely lose valuable time finding them. Both time and collectibles play a role in the final rank so when you do them ahead you can just concentrate on mastering combat.

Use Demon Evade! You gain more damage for a short period of time but now your attacks raise the rank much quicker. You basically have to perfectly evade (evade just as the enemy is about to hit you) while in demon mode. The keyboard input is E+Shift and direction which isn't very easy to press. I'm not sure if you can do this on a controller but on PC you can press 2 and Dante will keep his demon weapon out without the need to keep E pressed. When you do that you only have to evade at the right time without breaking your fingers.

Change up your attacks! This game loves it when you keep switching between different attacks and weapons and rewards you with a lot of points. That being said, if you master 5 maybe even just 3 different moves and keep switching between them the game will love you. If you combine that with Demon Evade a SSS rank is super easy. Remember though, more moves, more points.

Use Kablooey! This gun is more useful than it looks but you need to use it in a specific way. Make sure you have it fully upgraded so you can shoot 6 darts into enemies. Then fire all 6 darts into a single enemy. Do a demon evade and blow your shots up. This should give you an S rank from a single enemy. When the blast hits more enemies and you already build a decent rank this will instantly catapult you into a SSS rank.

Here are some good attacks that do nice amount of damage and raise score super quick:

Drive+Overdrive
Stinger+Trillion Stabs
Death Coil
Trinity Smash
Fully Charged Stomp
Buy In+Tornado

This is a personal note but maybe you have trouble with the attacks that require you to pause between attacks (I admit I used to click as much as possible). This game has a very good helping tool. Do the first part of the combo, wait and when you see your weapon flash finish the combo.