The original answer is quite brilliant and exciting, but in my pursuit for finding it, I've stumbled upon a rather strange alternative which seems to fit the riddle. Have a look at my intentionally forced theory and give me your opinion.
Sherlock Holmes speaking to Professor Moriarty near the Reichenbach falls
Why? Well, read on if you are interested.
I'm always in your thoughts, technically twice
If displayed correctly, just to be precise
Holmes was always a stinging thought in Moriarty's mind. The professor was directly involved in two cases titled "The Valley of Fear" and "The Final Problem". Hence Holmes would've crossed his mind twice.
If you see my left you will call me a con
and it would tear me apart if you called me John
Holmes have said Watson about this once. "If you would've seen my sinister (left) side, You'd know that I make a perfect villain (con)". Also, since Holmes is speaking to Moriarty we could assume the second sentence is about (Dr.) John (Watson).
Take down a note and drop your cup
All you need is what keeps up
The note might be referring to the one Holmes wrote to Watson moments before the death of his arch-enemy. Since the combat was done without any ammunition, Moriarty might be saying "All you need (for this final battle) is what keeps up (the strength of his body)
Please don't apply pressure or I will be off
and try to solve my riddle if you can take the scoff
Moriarty's final words perhaps? He might be inches away from slipping off the ledge. "Do not apply the pressure (spare me Holmes)". Since his right-hand, Sebastian Moran, was still living to complete his mission, the professor shouts "solve my riddle if you can take the scoff" (which is impossible, my organization is bigger than you think, Holmes...)