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
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
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.
Smash and Grab
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 all standard bio balls (marines, marauders and medics) and goliaths.
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, then you should get some bunkers and tanks down.
The key points are targeting, positioning and keeping your goliaths alive.
Targeting
-Always aim your goliaths at Void rays first, as they do much more damage over time.
-Use Marauders and goliaths to take down colossi quickly.
Positioning
Keep marines, medics and goliaths away from colossi. Colossi two shot marines/medics and you have to repair goliath damage. Use marauders to tank their damage instead.
Let your bio units soak up damage from air units while your goliaths use their superior range to take them down. Healing is free- repairing is expensive and micro intensive.
Keeping goliaths alive
Goliaths are super expensive compared to the bio units. Keep them near the back of your army and use them against air and colossi (they use their anti-air attack against colossi). If they get low on health send them to the nearest base where idling scvs can repair them. If you're going to lose a fight, leave the bio units behind to buy the goliaths time to escape.
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.
In Utter Darkness:
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:
The final mission, this one has you holding out for a pretty long period of time against waves of zerg.
[i]The key here is that you MUST kill the nydus worms around the map.[/i] 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.
