Not Updated For Current Season
This is my Karthas build which so far as been very successful. Allow me to explain the items, as that's the part that's always up for discussion hear; everything else stays the same.
Stat wise, mana regen is important because you are a mana hungry killing machine, but not so much that it's completely unmanageable. Blue golem buff helps, and a mana potion is great too if you find yourself spamming 'Q' a lot.
Karthas is the offensive juggernaut, and also the glass cannon. His item set up can vary slightly from game to game depending on the players you plat against. Mejais is nice if you're playing against squishy players who don't have a whole lot of HP, or if you're working with teammates that will help you get kills/assists, but is NOT a staple item.
Warmogs is fantastic for Karthas for many reasons. Firstly, he's very squishy, and the HP boost is phenominal. Secondly, he farms creeps better than any other hero in the game given that his aura is a toggle and his main attack is spammable (Q), and he can farm them while he's dead! Notice there are only 5 core items selected, leaving 1 extra item to be selected as per your preference or needs based on the team you are fighting against. Generally speaking, an extra defensive item is great such as a thornmail or banshee's veil. The other items selected speak for themselves.
Spell wise Flash and Teleport/Ghost will work. I prefer Teleport to ghost as it helps him keep his XP higher since he can return to the field of battle quickly, and it allows him the option to aid allies and gank enemies. Ghost is also a great option for getting away from a gank or suicide tower diving.
Overall, Karthas is a very practiced hero. If you can't lead people into your bombs, you won't do very well as him, and if you don't know how to toggle the aura you'll run out of mana too quickly and end up dieing. If you have 0 mana and 100% health, you might as well be dead because you can't defend your self. Go back and get mana so you can continue to reap havoc.
If you're dead, your DPS is 0 no matter how much AP you have... Well there's a small exception with Karthas, but you get the picture.