Malphite is considered to be either a tank, tanky-dps, or soak, depending on who you ask.
He is a primarily defensive champion that can build damage, but requires very high defense. This may be a little confusing to new players, because he has 3 abilities that improve his survivability (attack speed slow from ground slam, armor from brutal strikes, and passive health shield). The reason why Malphite is never viable as a straight dps or with a normal tanky-dps build is that all 3 of his defensive abilities require defensive stats to be very defensive. There are litarally 3 items that are REQUIRED on Malphite, and all three are defensive. So it does little good to try to build him as either a glass cannon or a regular tanky-dps.

Malphite is a tank in the sense that his primary responsibility is to soak damage and avoid needless deaths.

However, by Riot's classification of a tank, a tank must be able to:
*Initiate (Malphite's ultimate is arguably the best initiate in the entire game)
*Mitigate Damage (Any ability that mitigates damage and is not passive counts as damage mitigation for this requirement, Brutal Strikes provides bonus armor; therefore, it is damage mitigation).
*Disrupt the enemy team with sustainable damage or spammable cc. This is why Malphite is not considered a tank. His cooldowns are too long to be considered spammable cc. Attack speed slows are not even considered cc by Riot, so only his shard is even considered. Last but not least, Malphite's damage output is all burst, his sustained damage is not high enough or threatening enough to force the other team to move; therefore, Malphite is not considered to have the third and final requirement for a tank.

Malphite is first and foremost a soak. Soaks play the role of a tank in the sense that they absorb as much damage as possible, mitigate damage for themselves and the team, tank turrets when necessary, and be as disruptive as possible. But they do not do all of these things well enough to be considered tanks by Riot.

There is not champion that absolutely needs an item to be effective, but there are items that fit perfectly into that champion's kit so well that you'd be a fool not to take them.

For Malphite, they are RANDUIN'S OMEN, SUNFIRE CAPE, and FROZEN HEART.

Randuin's Omen: Gives necessary armor, health, and health regeneration. In addition, it slows targets that attack you and can be activated to further benefit Malphite's attack speed slowing capabilities. Works great with Ground Slam. The 5% cooldown reduction isn't much, but it helps.

Frozen Heart: Provides cooldown reduction, armor and mana. Allows Malphite to stay in the field much longer without blue pill or clarity. Armor is a necessity for ground slam, and Frozen Heart is the second most armor/gold efficient item in the game. The 20% cooldown reduction allows you to keep ground slam active for a larger percentage of the fight. The 20% attack speed slow aura stacks with ground slam and works well against ad carries.

Sunfire Cape: Primarily a farming tool for Tanks, Soaks, and Tanky-dps. Sunfire cape is perhaps the most commonly used item on tanky champions. It provides high amounts of health with medium armor, which are both great for tanks, and especially for Malphite. The aoe damage also helps out in team fights and for farming.

It is recommended that in all games with Malphite, regardless of the enemy team or optimal build order, you build all three of these items by the end of the game (assuming it is long enough).

Since Malphite is primarily a defensive champion. Defensive boots are best despite the advantages of the damage boots. In this section, I will list all of the viable boots on Malphite and why they are viable. Note, I do not consider Boots of Mobility or Boots of Speed, because ultimately you need something more than that.

Mercury Treads: These are the absolute best boots in the game. They give you tenacity and magic resistance which Malphite has trouble fitting into his build without this item.

Ninja Tabi: A not so good item that provides armor and dodge at a cheap cost. Generally you want to avoid these because there are usually disables and very few auto-attackers. You will also get lots of armor later, as much as 400 depending on the build, so these are really unnecessary.

Ionian Boots: These boots would be the best boots for Malphite if it weren't for Mercury Treads, they will usually get you to max cooldown reduction so that you can use your ultimate, ground slam, shard, and brutal strikes optimally. The problem is you really need the magic resistance and tenacity from Mercury Treads.

Berserker's Greaves/Sorceror's Boots: Not recommended, but usable. You will increase your damage output either way, but durability and utility are more viable.

You generally want a defensive book.
Straight maximized armor or magic resistance are good, but situational.
A balanced tank book like the one shown here is best. Notice that health/level runes are much better than the flat runes.

Masteries should also be primarily defensive or utility, I like a mix of the two, as it gives me more sustain and utility during the game than straight utility or straight defense. I don't consider enough of the abilities in either tree useful enough to take them down to the 21st point.

Spell Choices: I know flash is good, but honestly you won't need it on Malphite. Ghost will let you chase down enemies with flash quite well, and you shouldn't have trouble escaping with seismic shard. If you really can't escape with Ghost and Shard, Flash probably won't help you.

I use clarity because it improves staying power in lane and I often find it useful for replenishing allied mana late game. Very few people build enough mana for themselves, so this summoner spell becomes a godsend for most teams.

Atma's Impaler is a flat out under-priced and overpowered item.
700 gold for 45 armor, 830 gold for 18% crit, 825 gold for 2% of your health in damage. The damage part alone is less cost than a pickaxe, and by the earliest point you can get it it is better than a pick axe. With only 2.5 k health, and this build I have here would give you 2.8k, you get 50 damage, the same as a bf sword for half of the cost.

Reasons to get it on Malphite: Malphite needs to be able to survive well to maximize his damage and benefit to the team; therefore, most damage items are horrible on Malphite. Atma's Imapaler and Frozen Mallet are the two exceptions, you need magic resistance and the 3 required items + boots though, so Atma's is the best choice.
-45 armor improves the damage on ground slam by 22.5.
-18% crit-chance improves your damage by roughly 18%.
-Brutal Strikes causes your auto-attacks to deal 62% aoe damage and can increase the damage you deal by 40%, so the attack damage is very very good.

Malphite's kit does not work well with a true atma's build, but his kit does work well with Atma's Impaler.

As with all LoL builds, you will not usually get the entire build finished, so the pieces must be playable without the build being complete.

On Malphite it is most beneficial to start with a Ruby Crystal for Heart of Gold. Your passive can make up the difference that is normally made up by health regeneration.

Build Philostone second, usually it is best to start or finish this before getting boots to maximize the gold output. Keep this item until you are building an item where you absolutely need the space because you cannot buy the first starter piece to your next item in your inventory.

Boots are third, I recommend Mercury Treads.

Depending on the team, build Negatron Cloak for Force of Nature (best because you have limited space for Magic resistance items) or Chain Vest for Warden's Mail or Glacial Shroud. Glacial Shroud is usually better to rush unless you are against 2 ranged carries in lane.

Finish Frozen Heart before Omen, you might want to buy the health portion of Sunfire cape before doing this.

Finish Either Omen or Sunfire Cape next, this is your choice depending on how the game is going. After finishing one or both of the two, build a Force of Nature or an Atma's depending on the enemy team.

Finish Force of Nature or Atma's, if the other team really has no magic damage you can replace Force of Nature with a Guardian Angel or a second Randuin's Omen for health and additional armor, but Force of Nature has such good health regeneration it is still usually better to just build a Force of Nature.

You can start Atma's at any time, but there is no use building it if you're dying quickly.