The Reaper of Souls update has substantially modified this process.
There are 4 different machines that can be created which correspond to the particular fights to obtain a particular organ. No more opening 3 portals to guarantee seeing the uber you need.
The Infernal Machine plan can drop from any of the Keywardens and they all have a small chance to spawn a second key that is not particular to them.
To craft one of the 4 machines you only require the appropriate key and 100,000 gold coins.
The level 70 Hellfire Ring requires 4 organs, the new crafting plan which costs 5,000,000 gold coins from Squirt in Act 2 and 50,000 coins to craft.
The organs are:
It is still possible to obtain the original keys while you are level 60 and to craft the level 60 Hellfire Ring as per the legacy information that used to be contained in answers here (you can see that info from reviewing edits on the top answers).
It is not possible to stack the level 60 and 70 Hellfire Rings to obtain an 80% experience gain bonus.
It is not confirmed that the organs drop with the same rate as the keys. They pop only in Torment difficulties. The rates are listed in this answer.