Mass Effect 2

I'll be covering the two boss fights. I played through on Hardcore as a Vanguard so you might find tactics that are more effective for your build, but hopefully what I say will still be of some use. I'll keep things as spoiler free as I can by referring to the bosses as "it" or by the names they are called in the achievement descriptions.

You will need to switch to Hardcore or Insanity before you go to Illium to speak to Liara for the achievement.

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Soldier Tip:
"As a soldier I found bringing Garrus or Kasumi was really helpful to bring some balance to the team. You'll need some solid anti-tech."
Credit - Old Willy Pete
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Illium - The Assassin

Phase I:

This is a mostly long range battle where patience is the key. The Assassin will randomly teleport around the area and try to pick you off from a distance. Do your best to be aware of your surroundings and adjust your cover as needed. Return fire with a long range weapon whenever you have an opening. Your allies will likely be quickly picked off by The Assassin. I don't recommend reviving them during this part of the fight. They'll most likely just get taken out again and you'll want to save your medical supplies for the rest of the mission.

The Assassin will often send a shockwave your way. Whenever she does so, be prepared to leave your cover, get out of the way, and then return to your cover. If you do this fast enough you should take little to no damage. If you get hit by the shockwave The Assassin will land some hits on you and possibly kill you.

If you see The Assassin standing still while charging up biotic energy this means that you're about to get hit by a Charge attack. Switch to your shotgun and start strafing. Regardless of whether or not you get hit you'll probably be able to get in a hit or two before The Assassin teleports away again.

Phase II:

Once The Assassin takes enough damage you'll be left alone while some generic enemies and rocket drones come in. This would be a good time to revive your party so you don't get overwhelmed. Use Liara's biotics (and your own if you have them) to disrupt enemy humanoids whenever possible, and focus any shield disrupting skills you have on the rocket drones. Once these enemies have been defeated you will return to Phase I. The two phases will alternate until the end of the boss battle.

Hagalaz - The Shadow Broker

Phase I:

This fight starts out simple. The Shadow Broker is big, slow, and in the middle of an open room. Whatever squad member you chose will be unconscious for the entire battle, so it's just you and Liara. Just fire away at The Shadow Broker as he slowly approaches, using Liara's (and your own if you have them) biotics to stun it repeatedly. You should be able to take it down before it reaches your cover. If not, make a run for one of the pillars to your sides when it's not firing at you. Once its shield and barrier are down a cutscene will prompt you to run up to The Shadow Broker and melee it with B.

Phase II:

Phase II is much like Phase I, but with the added complication of the Shadow Broker having a large hand-held energy shield. I recommend standing behind a pillar (without using A to take cover) and slowly strafing clockwise around it as the boss follows you. This should leave a small opening between the pillar and the shield where you can get in some shots at its body without it shooting back. If it shoots at you speed up your strafing a bit. Continue to use Liara's singularity to stun The Shadow Broker when you get a chance. Not only will this slow its pursuit, it will also cause it to lower its shield and leave it wide open for attack.

As long as you don't take cover behind the pillar with the A button The Shadow Broker usually won't try to break through it. If he does, make your way to another pillar as fast as you can. I would recommend reviving Liara whenever she goes down because she provides a very useful distraction.

Once The Shadow Broker's shield and barrier go down you'll have a chance to punch it again. Phase II will then repeat. Once you punch it at the end of Phase II for the second time The Shadow Broker will go down for good and you'll get to see how this DLC ends.

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Engineer Tip:
"If you're an engineer the combat drone is very effective at distracting the Shadow Broker, just place it behind [it] and [it] will turn [its] back to you, making getting past the shield even easier."
Credit - Master Krane
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I hope this helps you. Feel free to comment with any questions or with any mistakes you see in this solution. I'll help out if I can.

I actually didn't change my difficulty to insane until after I spoke to Liara. I think you have to have it set correctly though before taking the cab to her apartment, cause from there on out you're locked into the mission until the very end. Another tip and I feel embarassed for admitting this but at some point Liara becomes a squadmate. and she kept dying all the time and I couldn't figure out why until I finally realized that I had not leveled her up. Doing so is obiviously a no brainer as it makes her stronger but also gives you more abilities to use. I just hadn't played the game in a while and didn't even think about leveling up when I started playing this DLC.

I was a soldier and only died once on Hardcore difficulty. All you really have to do is start off hiding behind the cover, and when the Broker smashes it, go behind the pillars as others have already stated. Seems like a no brainer since there is no other cover to be had, and he will destroy you if you stay wide open for long.

What I did was I hid behind a pillar, then got Liara to hit him with a Heavy Singularity (upgrade it to the max if possible). Then from there he was stuck, and I moved to a pillar on the other side of the room and fired disruptor sniper shots at him until his shields dropped, and I switched to incendiary shots to finish him off. I think I may have found a glitch because even after the Singularity expired, he wasn't moving at all. I only discovered that on the second phase of the battle (though the third and last time before you got to kill him). As long as you use Liara's Singularity, doesn't matter if he has a shield or not.