The Reaper of Souls update has substantially modified this process from the launch version.
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 machines spawn directly from the Keywardens. As of patch 2.3 each member of a party has an independent probability of generating a machine.
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.
The level 70 Hellfire Ring requires one of all four 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.
The key rates are reported to be but are not universally accepted as:
- Torment I: 25%
- Torment II: 28%
- Torment III: 33%
- Torment IV: 38%
- Torment V: 43%
- Torment VI: 50%
- Torment VII: 60%
- Torment VIII: 70%
- Torment IX: 80%
- Torment X: 90%
The organ drop rates are estimated to be:
- Torment I: 100% * 1 organ roll
- Torment II: 10% * 2 organ rolls
- Torment III: 20% * 2 organ rolls
- Torment IV: 30% * 2 organ rolls
- Torment V: 40% * 2 organ rolls
- Torment VI: 50% * 2 organ rolls
- Torment VII: 100% * 2 organ rolls
- Torment VIII: 10% * 3 organ rolls
- Torment IX: 25% * 3 organ rolls
- Torment X: 50% * 3 organ rolls