I highly recommend started the game on "Delta" for the first time, if you do not want to beat the game twice. It's not really difficult.

Some achievements may be bugged. You can try deleated saves and started a new game.
Saves are at %USERPROFILE%\Saved Games\CrysisRemastered. Delete the SaveGames and Profiles folders (make a backup first).
Thanks for the information to the user Muckelchen: https://gtm.steamproxy.vip/app/1715130/discussions/0/4943253385023029778/

A Little Trouble Parking
Discover the fate of the Lusca's Call




Easy Darlin'
Rescue the hostage




You Knew, Didn't You?
Regroup with Prophet upriver




Very Strange Readings
Infiltrate the excavation site




Livin' Up To Your Name
Board the VTOL for extraction




Delta: Act I
Complete 'Contact', 'Recovery' and 'Relic' on Delta difficulty




Pro-Aircraft
Destroy all AA guns around the harbor




Enjoy The Fireworks
Destroy the cruiser




Empty Platform
Secure the train station




One Careful Owner
Reach the end of 'Onslaught' in the tank you started with




You're On Your Own
Proceed to the mining complex




Going Underground
Enter the mines




It's On Like General Kyong
Defeat General Kyong




Delta: Act II
Complete 'Assault', 'Onslaught' and 'Awakening' on Delta difficulty




I'm Coming Home
Escape the mysterious structure under the mountain




Expedition Team
Escort Prophet to safety




I'm A Marine, Son!
Help the marines evacuate




Crysis Controlled
Complete the game on any difficulty




Cool In A Crysis
Complete the game on Hard or Delta difficulty




Strickland Would Be Proud
Defeat the flight deck invader




Close Encounter
Secure victory in the Battle of Lingshan




Delta: Act III
Complete 'Core', 'Paradise Lost', 'Exodus' and 'Reckoning' on Delta difficulty

To get these achievements, you need to perform secondary objectives. They are marked on the map with yellow dots.
Following Orders
Complete 3 Secondary Objectives




Without Question
Complete 6 Secondary Objectives




Perfect, Soldier!
Complete all Secondary Objectives

I got an achievement after clearing the platform at the "Awakening" level. If you didn't get it there, it may mean that you missed a mission somewhere.

Zoology
Pick up an animal

At the first level, pick up a turtle


Marathon Man
Speed sprint 3km

Run


Special Forces
Kill 200 enemies




Nano Ninja
Perform 5 consecutive kills without being spotted by an enemy
I advise you to get close to the enemies in invisibility mode, turn off the invis before firing and immediately turn it back on. This way you will not waste all the energy of the suit and at the same time remain unnoticed.

Long Distance Relationship
Kill an enemy 200m away

Easy to get at the beginning of "Assalut". Go to the Commander at the Hilltop, take a sniper rifle and shoot to the korean in boat.

Tank Buster
Destroy 5 enemy tanks

Easy to get at "Onslaught".


No Fly Zone
Destroy 5 helicopters

Just use a Rocket Launcher.


Keen Observer
Tag 30 enemies using the binoculars

Click on B and mark all the Koreans along the way.


Something For Every Occasion
Use all weapon attachments

• Flashlight
  
• Laser sight
  
• Silencer
  
• Open sight
  
• Reflex sight
  
• Assault scope
  
• Sniper scope
  
• Tactical attachment
  
• Grenade launcher

Team Raptor
Kill 400 enemies




Choke Hold
Kill 20 enemies with grab

I recommend this way to kill aliens on the ship. At the moment when they attack you, press F and one-shot the enemy

Weapons Master
Perform a kill with every firearm




This Is My Rifle
Customize a weapon to use all 5 modification points

Don't forget about incendiary ammo. It can be found on the alien ship.


Catch This!
Kill 10 enemies by throwing an object at them

You can first knock out the enemy, and then throw something at him.


Knock-off Knockout
Kill an enemy Nanosuit soldier with a Strength punch

The easiest way is kill a nano soldier with a machine gun, they are not particularly agile.