KARTHUS is a very strong mid laner because of his crazy sustained damage, has one of the best ultis in the game, one of the best passives in the game, and is one of the best farmers mid lane. Now this doesn't matter if you don't know how to play Karthus or if your R key on your keyboard is broken.

For Reds I take Magic Pen. these are essential on all AP's no real alternatives here.
For Yellows I take Flat Mana Regen. so in lane i always have mana to farm with my Q because you just cant simply farm with your auto attks. Alternatively you can take HP per lvl or Mana Regen per lvl, but you'll have to be more sparing with your Q in lane cause your E passive isn't enough to keep you up to full all the time.
For Blues i take AP per lvl because the AP late is very nice. Alternatively you can take flat MR or per lvl if you are going up against a very harass heavy mid (Kass,Ryze,Cass,etc.)
For Quints i take Flat AP because the extra early AP is very nice for multiple things. Alternatively you can take Movement Speed Quints which are very strong on Karthus.

21 0 9
This aims at dealing lots of Magic damage with some utility. One thing to note about these masteries is in Off. tree i take 3 pts in brute force instead of Mental Force because anyway you look at it 3atk dmg is better than 3ap. Nothing more really to say here pretty standard.

This build is very standard for Karthus. Pretty much start boots 3, rush catalyst and Sorc Shoes, and that right there is your core build. After get your ROA immediately into Deathcap, after DCAP theres 3 options you can go depending on the other team firstly you can go WOTA which is what you would get if your dying to sustained dmg over time. If you are getting bursted down immediately get your Zhonyas. And lastly if they have alot of MR early get your Void staff. But no matter what your end build should look like the one above. One thing to note is the item Archangels now honestly Im just not a big fan of this item on Karthus, or at least the way I play him I think this item can work on him well but i just don't know where to sub it in for because all of the above items are so good on Karthus and they mesh really well. I have seen rushing a ToG on Karthus to be quite strong early, this item is quite good on him but as i said i prefer cata into RoA. But if you are going to get a Archangels don't make it your first item, get Tear of The Goddess and finish Arch as 5th or 6th item. One more note always when you back to base buy wards and Hp pots these are just so amazing early in the laning phase and wards you should always be buying and placing to protect from ganks or to steal buffs or to kill someone off doing something with low health just endless uses to having map vision.

Max Q immediately pretty obv., after get your E taking 1 pt in W at 4. Now One thing i need to note about your Q make sure you farm with Q dont even try to last hit with Auto, it just wont work he has one of the most messed up Autos in the game, but it doesnt matter because of Q plus E passive plus runes and Mast you can always have enough mana to last hit with Q. On the topic of mana make sure you tell your jungler to give you blue it is so imperative on Karthus the Cd reduction so you can Press R more and more the inf mana so you can farm inf.

I take flash exhaust. Why? Because Ignite on Karthus just really isnt needed with Exhaust you can 1v1 practically any1 in lane as long as you pick your fights and dont get greedy.
If you want to take Ignite on Karthus its not the end of the world it works ok, but i strongly recommend Exhaust.

So basically Karthus is a really strong pick for mid lane. Has one of the best ults in the game as long as your R key isnt broken. But just make sure you farm hella hard early and mid game and destroy late because Karthus scales amazingly into late game and even when hes dead hes alive.