I tend to play Malph on solo top, or on bottom with a good partner. My build is usually something like this:
1. Start with Doran's Shield. This usually makes me beefy enough to where I can hang around until at least 6 without having to B. 2. Ninja Tabi. 3. Iceborn Gauntlet (starting with Glacial Shroud) 4. Sunfire Cape 5. From here, if my team is a bit on the squishy side, I'll tend to go for Warmog's. Otherwise, Abyssal Scepter or Liandry's Torment are good choices. I tend to play with my friend who uses characters with movement inhibiting spells a lot, so LT's burn effect does extra damage (look up the item... it's pretty nifty). 6. By this point, the game is usually about over. If it goes on for a long time, I usually end up with some combination of the above items. I pretty much always go for the first four steps exactly everytime though. It's pretty rare that I have a bad game with Malph.
Keep in mind that Malph's Q and R gain extra damage through AP, while his E actually gains extra damage through your defense. So extra armor on Malph is never a bad thing. With this build that I typically stick to, I rarely have issues not doing enough damage with Malphite. Against tankier characters, yeah, you might have issues, but Malphite trumps most champions 1v1 still. Also, when initiating team fights... try to aim your R for where it will hit the most champions at once (getting several hits with his R followed up by his E can be devastating and make the enemy team IMMEDIATELY start running).
As far as "what to do", the other posters are right. You want to be able to combo them to death by 6. This is easy as hell laning against squishy characters. Nobody really seems to ever expect Malphite's damage, so usually if you can poke them down to 2/3ish HP (maybe not even that much depending on the character), then you can usually go R to clear the gap and knock them up, then E for the shockwave, Q for when they start running away, and then W to finish them off. If SOMEHOW they get away, then Ignite or Q again since the cooldown is pretty short.
If you're feeling ballsy pre-6 and your enemy isn't playing very defensively and is squishy (and they aren't hanging too close to their turret), then once you get Malph's Q and E you can often hit them with Q, clear the gap with the movement boost, and hit them for more damage with your E. Sometimes I can manage an early pre-6 kill this way. At the very least, you can wound them pretty badly and maybe force them to use a summoner spell or have to B early, or set them up for an early gank by your jungler.