Regarding Farming:

Farm with your Q and only, after the 10-minute-mark. Its cooldown, even with blue buff, is perfect for the Tear to stack as quick as possible.
To clear out huge creep waves, the skill sequence goes like this:
Q -> R -> E -> Q -> W -> Q (and repeat)


Regarding Items:

-You should be able to get Rod of Ages by the 16-minute-mark. Otherwise, it's useless. Your RoA will provide the tankiness you will need to easily 3v1 early-to-mid game.
-The second Tear of the Goddess should be bought AFTER you have bought Manamune/Muramana. Otherwise it will not stack by the time you buy Manamune. That is because the passives are unique, when it comes to stacking mana. And, if you don't get a different passive, they are just gonna stack the same amount and after you've upgraded to Manamune, the second Tear of the Goddess will reset itself. So do that.
Note: DO NOT GET ARCHANGEL'S STAFF FIRST. ITS PASSIVE IS THE SAME AS TEAR'S PASSIVE WHICH MEANS THAT YOU WILL NOT BE ABLE TO STACK A SECOND TEAR OF THE GODDESS UNLESS YOU GET ARCHANGEL'S UPGRADED TO SERAPH'S EMBRACE.
-I personally do not worry that much about Ryze getting a Seraph's Embrace. That is cause you will literally be able to 4-hit everyone after you've upgraded your Manamune to Muramana.
-Your Muramana should be existant in your inventory by the 30-minute-mark.
Note: Since you are farming like hell to stack up that Manamune, you don't need all the gold that you can take. You will be full items really early (around the 35 - 40 minute mark) and after that, all the gold you get will be wasted. So, you do not have to take all the kills, leave your mates some gold, they deserve it for helping you and for letting you stack your Manamune. As long as you get your Rod of Ages by the 16 minute mark, it is ok to let all the kills slide and just focus on minion farm.

When PvPing, keep in mind that your passive reduces all your cooldowns except for the Spells' cooldowns by 1 second everytime you cast a skill. Keeping that in mind and, keeping in mind that Overload has a really low cooldown, you should be able to follow some certain skill sequences, depending on your CDR.
-Early game: Keeping in mind that you have maxed out your Q and its cooldown is 3 seconds, (2.94 to be precise) your skill sequence in PvP should be:
A.Q -> E -> W -> Q -> R -> Q (and repeat)
OR
B.Q -> R -> E -> Q -> W -> Q (and repeat)
OR
C.Q -> W -> E -> Q -> R -> Q (and repeat)
-Mid-Late game: Your Q cooldown is 2.1 seconds and that is really convenient since you can now spam Q in-between the other skills. So, your skill sequence should be:
1.Q -> R -> Q -> E -> Q -> W -> Q (and repeat)
OR
2.Q -> E -> Q -> R -> Q -> W -> Q (and repeat)
OR
3.Q -> W -> Q -> E -> Q -> R -> Q (and repeat)
OR
4.Q -> R -> Q -> W -> Q -> E -> Q (and repeat)
OR
5.Q -> E -> Q -> W -> Q -> R -> Q (and repeat)
OR
6.Q -> W -> Q -> R -> Q -> E -> Q (and repeat)

Q. Why so many skills sequencies?
A. Cause for a different PvP situation you encounter, there should be a skill sequence that's best for it. For example, for a big creep wave, you should follow skill sequence (B) for early game and skill sequence (1) for mid-late game. Another example, for 1v1 you should follow skill sequence (C) for early game and skill sequence (3) or (6) for mid-late game.