Simply finish the game, you can't miss these as they are story related and achieved simply from progress:

• Heavy Burden - You infiltrated the mansion and assassinated Diana Burnwood
  
• Kingslayer - You fulfilled Birdie's contract and eliminated The King of Chinatown
  
• Chamber of Secrets - You located the hotel room number 899
  
• Catch a Ride - You boarded the train and escaped the Chicago PD
  
• Forepost - You assassinated Wade's men in Chinatown
  
• All Bark and no Bite - You assassinated Wade
  
• The Bartender Always Knows - You questioned the bartender
  
• Signature Weapon - ou re-gained your signature Silverballers
  
• Like Stealing Candy From a Baby - You circumvented Lenny's crew and secured Lenny
  
• Step Into The Light - You found your way through the mines
  
• A Heavy Blow - You assassinated the facility leaders
  
• The Killing Fields - You eliminated the Saints
  
• Jailbird - You infiltrated the courthouse and accessed the jail
  
• Hour of Reckoning - You caught up with Skurky
  
• True Form - You visited Tommy the Tailor and acquired a new suit and gloves
  
• Destroying Something Beautiful - You assassinated Dexter's assistant, Layla
  
• The Final Countdown - You assassinated Blake Dexter
  
• A Personal Contract - You eliminated Travis and fulfilled Diana's contract

If you follow the story these are easy to accomplish, but due to optional choices may be missed on the first play through:

• Damage Control ~ In the King of Chinatown, instead of playing stealth, cause chaos and then kill all the hostile enemies to regain control of the situation. You basically have to get spotted and then kill any alerted enemies to calm the situation again.
  
• Faith Can Move Mountains ~ In Fight Night, choose to fight Sanchez and defeat him with your bare hands instead of any of the numerous other ways you could kill him.
  
• Not Worth It ~ In the mission End of the Road, leave Lenny in the desert alone instead of killing him.

These are granted for doing specific actions or repeated actions.

• Take me to the geek - run a benchmark mode (in the Options menu in the launcher)
  
• Under Wraps - simply hide a body
  
• Sandman - subdue a person
  
• It's All in the Wrist - throw anything sharp into someone's head
  
• Rocksteady - execute 3 hostiles using Point Shooting
  
• One of the Guys - Blend in with a disguise and fool someone
  
• Inconspicuous - avoid getting detected throughout a level
  
• True Potential - unlock a technique
  
• The Russian Hare - perform 47 headshots with a sniper rifle
  
• Whoops - kill someone with an accident (like falling pile of bricks)
  
• One With the Shadows - evade the enemies (just get detected and then hide somewhere)
  
• A Taste for the Game - complete 10 challenges
  
• Reach for the Stars - complete 50 challenges
  
• Grand Master - complete 100 challenges
  
• Absolution - complete the game on Professional (or higher) difficulty
  
• Silent Assassin - Achieve Silent Assassin rating on any mission
  
• Information is Power - recover all of the evidence tapes
  
• Jack of all trades - collect all 20 play styles

For specific instructions on Evidence locations or collecting play styles, guides already exist:
https://steamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=184554659
https://steamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=1790140393
There are also tons of guides for in-game challenges, here's just one:
https://steamcommunity.yuanyoumao.com/sharedfiles/filedetails/?id=168863067&searchtext=challenges
Aside from the "quantity" to achieve Grand Master (100) there's no specific challenges required for Steam achievements.

CAN NOT CURRENTLY BE COMPLETED BY PLAYING ~ unless you use mods or a server emulator*

These achievements are completed through activity in the Contracts Mode.

• Blood Money - You completed the Contract Basics Tutorial
  
• First Contract - You completed the Creating Contracts Tutorial
  
• Thumbs Up - You liked a contract
  
• Partners in Crime - You played a contract made by a friend
  
• Competitive Spirit - You created a contract competition
  
• Contender - You participated in a contract competition
  
• Set for Life - You earned 1 million contracts dollars
  
• Self-improvement - You bought an upgrade for a weapon

Contracts mode is currently (and likely to remain) disabled due to choices made by the former publisher Square-Enix in response to the GDPR law changes.

The official announcement from the developer is here:
Hitman Absolution Service Message[www.ioi.dk]

What they are too polite to tell you is that the problem is with their former publisher Square-Enix (SE). At the time Absolution was released in 2012, ioi (the developer) was an owned subsidiary of SE (the publisher). Due to various business decisions ioi split from SE to become independent in 2017. The contracts mode for Absolution used various SE licensed IP objects from other SE games (such as Tomb Raider, Deus Ex, etc) and because of this, SE chose to keep the contracts servers when ioi parted from their company.

In 2018 the European Union released new privacy laws known as the GDPR which demanded sweeping changes to how all servers and websites manage data and information. These changes required extensive back end changes of all existing server infrastructure in the world. Many games and websites chose to shut down rather than spend the manpower and resources required to comply with the new laws. SE chose to shut down the Absolution Contracts Mode servers in response to the GDPR.

Can they be fixed and brought back online? In theory, yes. As a note: ioi owns, controls and updated the Contracts Mode servers that they have under their control for the newer games, that includes HITMAN (2016), HITMAN 2 (2018) and HITMAN 3 (2021). The newer game servers are in full compliance of the GDPR.

The problem for Absolution is that ioi does not have access or control over the Contracts servers. Because Absolution's Contracts Mode contained SE IP property, (for other games) SE chose to not release the servers to ioi during the split of the companies. And because of the GDPR SE was unwilling to spend resources updating the service themselves.

That's the limbo currently in place. Until and unless SE are willing to either bring the servers into compliance with the GDPR they can not be brought back online. And the alternative for ioi to fix them would require SE to hand over the servers for them to manage, which would require negotiating the use of other SE licensed game material that is no longer available to ioi now that the companies are not legally combined.

There are a variety of cheat methods and mods that can be used to complete these. The only one I will list is this:
Absolution Server Emulator[github.com]
If you need instructions for using it, you'll need to contact the person that made it through GitHub or on his Discord: https://discord.com/invite/5WW3C3G2M3