I am he who walks above the abyss
upon a mountain of sky
I am he who descends to the blackest nights
to the ever-slumbering restless
I am he who releases his star
which will shine and burn
I am the abyss / I am the storm / I am the sky
My strategy:
Kick major ass.
Naw just kidding.
Seriously though, in any map I play, I either start off by getting crap loads of headshots with the Pistol, or if I plan on grabbing a sniper, I kill solely with my SMG so I can make the bullets worth something (I save my bullets for the M6 and don't make a waste of the SMG I'm going to drop for the Sniper).
As it moves on, brutes will either be taken down by sniper fire, SMG fire, and then a headshot, or a plasma weapon and then a headshot.
Squads of Brutes with grunt squadmates are taken out like this:
A. I stick the brute and make the grunts either run or try to do a suicide run on me.
B. I take out the grunts first which makes less guns in the squad able to fire at me (can't fire if you're dead), plus, it also means that if the Catch skull is on I won't get stuck. This leaves me to focus solely on the Brute.
For Brute Chieftains I always try to grab a plasma pistol. If not, I will shoot my SMG or another weapon a little bit at a time until their shields drop. When they do, I try and do as little body damage as possible. The pistol does hardly any damage, so it will not set off their invincibility unless I've shot way too many bullets. As soon as the shields are down, I headshot them with the pistol until their helmet pops off, giving me an easy headshot kill.
Buggers suck, because I usually do not carry my SMG, and that is the most effective weapon to kill them with. Hopefully I am by a turret so I can rip it off and use that third weapon to mow them down as quickly as possible.
Before each third round, when Black Eye turns on, I make sure to grab a gravity hammer to use in the fifth wave of the second round. If I'm sure to use it sparingly, it will last most of the round and save me when my stamina is low, allowing me to survive for much longer (even if you hit with the shockwave, it still counts as a melee hit, therefore your stamina regenerates.)
When tilt turns on, the shielded buggers get really hard, but under sustained fire they drop easily enough. Swarms of buggers will still be something to worry about though, as they always are. Plasma Pistol or Carbine is my preferred secondary weapon, next to the Pistol. The pistol is great for killing the small infantry and for finishing off brutes. It's just not good for killing buggers, especially the shielded ones, or Hunters. I try to keep one on my person at all times.
Good strategy for facing a Brute Chieftain with a weapon that is no good: if you come face to face with him and can't find a weapon that will kill him fast enough, get into an open area and make him lunge at you. He will lunge straight at you, allowing you to walk sideways and forward. This places you behind and to the side of him, making his attack hit air or a wall.
That's all I can think of right now, that pertains to my general battle tactics in Firefight. There's more to it, but that's how I approach most engagements.