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A work in progress. You have tips/tricks/etc, feel free to post them here. If I messed up somehow, don't hesitate to correct me.

How's That For Starters?(10 G) - Complete dataDyne Central: Defection.
Simple enough. Just finish Mission 1 on any difficulty. (meaning you have to complete all three parts.)

Both Barrels(5 G) - Kill an enemy while dual-wielding.
You can get this in Mission 1: dataDyne Defection by going to all the floors you can via the stairs and killing the dataDyne Shock Trooper on the last floor you can visit before having to go on the elevator. Or you can load up a Combat Simulation game and have a loadout that contains at least a gun that can be dual wielded. (I suggest using the standard Pistol set on the Facility/Felicity map and have as many opposing MeatSims as possible. Since the MeatSims are easy to kill, Facility/Felicity is confining enough to find a dual set as well as it keeps the Sims not too far from you.)

Act Your Age, Joanna(5 G) - Destroy Carrington's wine collection.
To get this achievement, durring Mission 2: Carrington Villa when you gain access to the interior of the Villa, when you're in the basement, go left. There you will find Carrington's wine cellar, as well as a few dataDyne troops. Kill them and shoot/melee all the bottles. When you hear Carrington say "Act your age Joanna.", you know that you've destroyed the wine collection and gained an achievement.

Golden Days(10 G) - Complete a Combat Simulator game on Felicity, Complex or Temple using Classic Weapons.
Yet another easy one. I recommend going w/ Felicity and the Classic Pistol set.

Double 64(10 G) - Kill 128 enemies using secondary fire mode.
This one can be a bit time consuming, but fun. I got this one via the multiplayer by playing on the Complex map with the Dragon as part of the weapon set. I found a good area that overlooked two areas that met as well as giving me a good view of an opposite overwatch. I set the Dragon to it's secondary mode and just planted them over and over again in three or so spots where the enemy often came after me. They often carried Dragons, so I didn't have to worry about running out of mines. (just be careful where you put them. They will blow you up too!)

phillipo suggest that you should play 'Hold the Briefcase' and stake out one room. The sims will come at you constantly because you have the case. His weapons of choice for this achievement? The AR34 and Laptop Gun. (he also recommends turning on 'One Shot Kills' to make this go by faster/easier.)

Camera Shy(10 G) - Destroy 5 security cameras.
There's should be five cameras at the beginning of Mission 1: dataDyne Defection. Shoot them until they blow up.

Deadly Laptop(10 G) - Kill 50 enemies using the laptop in sentry gun mode.
Easiest way to get this one is through multiplayer. Set up your weapon set to have the Laptop Gun, pick a map, have as many Sims as you can (have people too if you wish, but they can be smart and destroy the gun), pick up a Laptop Gun and set it to secondary mode. Throw it at a nice spot and wait for your enemies to come on by. You will have to do this several times, as the Laptop will no longer work once the clip is spent.

Paci-Fist(10 G) - Complete any Solo Mission on Special Agent or harder using only your fists.
My guess is that Mission 1 would be best for this, but I don't know what the other objectives are on the harder difficulties, so I don't know if weapons are needed then. Please correct me if Mission 1 isn't the right choice here.

Tools Of The Trade(10 G) - Get a Bronze rating or better with all 32 weapons in the Firing Range.
With this one, all you really need to do is beat the whole game at least once (in order to unlock all the weapons for the shooting range) and get a high enough score with them in order to obtain at least a bronze medal for each weapon. Some exploring is also necessary to obtain some weapons, such as the Slayer.

Versatile(10 G) - Complete a Combat Simulator game using each of the six Scenarios.
Just play (and possibly win) a game in each mode of the combat simulator. (thanks phillipo!)

Prime Target(10 G) - Complete every Challenge up to and including 29, the 10th Prime number.
Sounds like all you have to do is complete all the challenges.

A Friend Indeed(10 G) - Complete any mission in Co-operative Mode.
This means the whole thing, not just one part. (For example, if you were to try to get this on Mission 1, you'd have to complete all three parts to get the achievement.)

...Who Needs Enemies?(10 G) - Complete any mission in Counter-Operative Mode.
Obtained much the same was as A Friend Indeed, except instead of working cooperatively, you're trying to kill one another.

dataDyne Specialist(10 G) - Kill at least one enemy with each of the dataDyne weapons.
dataDyne weapons:
-CMP-150
-MagSec
-Dragon
-Super Dragon
-Combat Knife
-Shotgun
-DY357 Magnum
-K7 Avenger
-Devastator
-DY357-LX

Carrington Institute Specialist(10 G) - Kill at least one enemy with each of the Carrington Institute weapons.
Carrington weapons are:
-Falcon (all variants)
-AR34
-Crossbow
-Sniper Rifle
-RC-P120
-Laptop Gun (possibly)
-Laser

Maian Specialist(10 G) - Kill at least one enemy with each of the Maian weapons.
Maian weapons are:
-Callisto NTG
-Phoenix
-FarSight XR-20

Agent(10 G) - Complete the Solo Missions on Agent
Beat the entire game on Agent difficulty.

Special Agent(15 G) - Complete the Solo Missions on Special Agent.
Beat the entire game on Special Agent difficulty.

Perfect Agent(20 G) - Complete the Solo Missions on Perfect Agent.
Beat the entire game on Perfect Agent difficulty.

Crowning Glory(5 G) - Earn all the Leaderboard Crowns.
There are 26 different 'challenges' that need to be completed along with five unlockables. Further details & explanations a few posts down.

That's all 20 achievements for you. If Achievement Hunter (aka Rooster Teeth) does any videos for these or if I find a good vid of anyone explaining/showing how to get it/them, I'll post them here.

Next up, avatar awards!

Getting The DY357-LX
Probably the hardest gun to get in-game. There's only two people who carry one: Trent Easton and a guard in the second bonus mission Maian SOS. I haven't had a chance to play any of the bonus missions, so I'll just explain how to get it from Easton.

There is only two missions where you can directly encounter Easton: Mission 5: Air Force One and Mission 5: Crash Site. In Air Force One, you can't get close enough to him so don't even bother getting it from him there. So that leaves Crash Site. Easton is hiding in a cave with the President (so far in all my tries, it's the real President and not the clone) surrounded by security bots. You'll know which cave he's in by a few details:
1) It has enough light so you don't need the Night Vision
2)There is two 'levels' to the cave
3)Unlike the other caves that are covered in snow, this one is primarily brown.

You have to approach Easton on the lowest level of the cave. If you try to get to him from above, he'll run away before you can even get close. To get close to Easton, you have to quickly take out the 'bots. Once you get that done, quickly switch from your weapon to Unarmed secondary. Run up to him and hit him enough times to disarm him, granting you the uber powerful DY357-LX! (just remember that when you obtain the golden magnum it doesn't have a whole lot of ammo with it.)

Happy hunting!

Yet another reason why Crash Site is your best chance to get it.

A more detailed explanation on how to get those illusive crowns:
http://www.mundorare.com/games/perfect-dark-xbla/crown-guide