I've just beaten the entire campaign on brutal difficulty (including the alternate branches) and figure I'd share tips from some of the harder missions in case anyone is having trouble.
Everything is protected by a spoiler tag, so you can avoid the ones you don't want to see.
The Dig
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This mission sucks. The key is using your laser cutter to kill enemies. Target templar->colossi->immortals in that order. You might need to target archons too in the early game. Always target the first wave that comes out first, as waves that come out after it pause for longer before attacking. DO NOT WASTE BEAMS ON REGULAR UNITS. You'll get stuck having your beam shoot a ton of stalkers when it could be speeding up the map or hitting more valuable units. All the regular ground guys die to siege fire anyway.
For the southeast, I walled in the area north of the expansion with planetary fortresses, siege tanks and scvs. The area to the northwest is much harder- bunkers and siege tanks were the best I could come up with. Remember to manually target templar that avoid your beam with your siege tanks.
I tried breaking the protoss with attacks to get the building kill achievement, but it proved far too difficult. Just wall in.
The Great Train Robbery
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The key here is the diamondbacks. You get a few to start, and you should immediately seek out the other diamondbacks on the map with them (they're fairly easy to find, as they reside on small plateaus around the sides of the map).
With your main ground army that isn't a diamondback, go NW of your base and clear the expo. You'll need this soon enough.
The first few trains can be handled by diamondbacks alone.
The next few trains you can bunker up at the and hold down with a bio army and diamondbacks at the intersections of the North/Middle tracks and the Middle/South tracks.
Once the hellion death squads show up, salvage your bunkers and retreat into your base. Go after the train whenever a patrol isn't about to run into it. You will most likely need to focus fire the train exclusively, ignoring the guards, in order to kill the train before you run into another patrol. Make sure to pick up the salvage afterwards.
This mission could be more or less hard depending on what units you have available. I did this with only diamondbacks and bio (no tanks or air) but it was pretty tough.
Welcome to the Jungle
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In this mission you have two main threats: Colossi and Void Rays. The scouts do poor damage against ground, and their standard gateway/immortal units aren't anything that bio balls can't deal with.
There's no special trick or super useful unit for this mission, at least not when I played. I did it with straight MMM. You probably want 3x as many marines as marauders, as there is quite a bit of air in this mission and marauders have much longer life expectancies than marines due to the colossi.
When you're starting out at your first base, don't bother with bunkering up and defending. Just get a critical mass of bio units (MMM) and then push out. Stop only the seals that are near your base. Give them the ones that are on the other side of the map.
After you take the middle base you should get some bunkers and missile turrets.
The key points are targeting and positioning.
Targeting
-Always aim your marines at Void rays first, as they do much more damage over time.
-Use Marauders to take down colossi quickly.
Positioning
Keep marines/meds away from colossi. Colossi two shot marines/medics. Use marauders to tank their damage instead then come in with the marines.
Don't forget to expand to the middle position as well. That should be your main base of operation with turrets, bunkers and 3-4 scvs that are just idling that will autorepair injured mech units coming back.
Supernova
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Banshees are the key here. Remember that. Also, you only need about 14 scvs. Probably less.
At the beginning, build a tech lab for your barracks and your starport. Lift off the factory- you won't need it right now. Build MMM (stick to mostly marines until you pick up gas later, as you need gas for banshees) out of the barracks and banshees out of the starport. Don't bother with bunkers or anything.
Take your initial banshees and clear a path east. You can kill all the units while cloaked, including the closest protoss relic. When you have a couple more banshees joining up, start poking northeast towards the photon cannons. Get an angle where you can kill a cannon and then the pylon, then finish off everything else while cloaked. Rotate injured banshees backwards, and go repair if you really need to. Send your MMM force in to go pick up the resources and relics left behind.
Keep in mind your main base will be attacked 3 times. Once by zealots, and twice by mixed gateway units. Make sure your MMM force is around to defend that, preferably with banshees too.
When you need to, pick up and move to that previously cleared small patch. Don't bother building tech labs for your barracks- just land and build marines. Don't land the starport or factory. Don't panic and lift too early. The fire moves REALLY slowly. You can pick up and move away safely even when the fire is already on the same screen as the buildings. Get whatever mercs you can before the fire hits- MMM units, siege tanks or banshees are preferred.
Now you should have ~7 banshees, and a decent MMM force. Use your banshees to clear the top of the ramp (again, don't be afraid to attack cannons, as you two shot them with your banshees). Get your MMM force up the ramp, then use banshees to draw units towards the MMM. Take out pylons asap- the enemy will warp in constantly. Drop the factory here and put down a tech lab. Build ~3 siege tanks. Don't forget to move the CC here when the other mineral patch runs out.
Repeat the same thing for the next area. Use banshees to clear anything outside the base, even photon cannons. Keep the banshees alive. When you have the relic and the outside of the base is clear, use your siege tanks with MMM support and banshee sight to lure guys out of the base and kill the cannons. Charge and take your final base.
This is where it gets tricky. Get your factory, barracks and starport landed here and tech labbed. You should hopefully have plenty of resources from your banshees + MMM raiding earlier. Use siege tanks and marauders to get the closest relic (marines + psi storm = fail). Keep cranking banshees, siege tanks and marauders out of your buildings.
On your final approach, use one siege tank and banshees in the front to lure with MMM and siege tanks in the back. After you get the guys at the very top, you'll need to rush marauders (and only marauders) up the ramp to snipe the templar up there, then run straight back down to the tanks and get healed. After the templar is dead, rush everything up the ramp and hopefully win it. Your cloak is kinda useless here because there is an obs, but if you have scan/science vessel/raven you can snipe the obs and get more out of your banshees.
Engine of Destruction
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Loki is on the east side of the map, roughly below the middle. It's at the 4th base you destroy (don't really need to go out of your way to kill it). It's a battlecruiser that will lift up when you near it. Don't trigger it until you have some serious AA. It has 500 hp and 4 armor, and fires its yamato cannon freely in addition to some sort of ground barrage.
In Utter Darkness:
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This is the final defense scenario with protoss. For starters, get tons of colossi and keep your phoenixes alive. Colossi are good units in general, and you need the phoenixes to kill the brood lords that harass you occasionally.
In order to survive, WALL OFF YOUR BRIDGES WITH DARK TEMPLAR. They don't get detectors for about 5 minutes, then get them infrequently, then often. When they do get detectors, run in your flyers to snipe the overseer, then rebuild the dark templar immediately. Once you get to really late waves, wall in with a dark templar wall and a zealot wall right in front of them. The zealots will buy you time to kill the overseers before the reach your DTs.
When your lines start falling, withdraw to the inner base where your mothership spawns. Hopefully you have enough carriers to do some serious damage. Fly to the southwest corner of the map (the enemy units will congregate there) where not all the ground units can hit you at once, and the overseers take longer to get to you. Keep killing units, sniping overseers as they come.
If you're having a lot of trouble getting those last kills, spam right click on all the zerglings, who will die much faster than other units.
All in:
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The final mission, this one has you holding out for a pretty long period of time against waves of zerg.
The key here is that you MUST kill the nydus worms around the map. They appear with boxes around them on your minimap and spawn orange colored zerg instead of the standard purple. You CANNOT hold out against them- I tried, really, really hard with siege tanks, bunkers, scvs, planetary fortresses, nukes... not possible. Thankfully, killing them isn't that hard. I did it with Battlecruisers, but banshees are probably better.
When Kerrigan comes out, she has 1500 hp and some sort of weird unlabeled damage resistance. Battlecruisers are the most effective unit against her, but she'll auto kill a few of them in every fight.
I suggest bunker walls with MANY scvs repairing and siege tanks for support.
When you use your final artifact blast, get all your units onto the high ground with the artifact. Before this happens cover the platform with supply depots and wall off the entrances with barracks (you will probably have plenty of excess minerals). This will prevent mass nydus worms from appearing on the plateau in the closing seconds, which will otherwise kill you.
If you have a problem with a mission I didn't list here, post it and what issues you are having and I'll add my suggestions.
BTW: Beat all 29 missions (25 campaign missions, +1 secret mission, +3 "branching" missions that were the option missions you couldn't do on the first runthrough) and you get this:
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