On frequency, if control is responding to the distressed aircraft you should stay quiet and let them work the problem. This would not be a great time to ask for flight following or traffic advisories and you should expect neither. You should be listening for instructions broadcast to all aircraft in the area and possibly an announcement to change frequencies.
A controller may try and hand some aircraft off if possible to ease their load.
Depending on your situation, and if possible, it may be warranted for you to fly out of the area, or avoid it if you are inbound. In some cases ATC will work to get the aircraft to stay in an altitude block if you can hear this assignment, you should stay clear of that block.
It's not really your responsibility to play chase plane but reporting things on frequency may be warranted if the channel is free, you are not stepping on a controllers attempt to contact anyone, and the information is pertinent to the situation. Perhaps you saw the aircraft crash site from the air; if you can provide a definitive location it may warrant reporting. Saying something like, "Hey I see that guy" is not really helpful especially considering today's radar (although coverage is far from perfect) and soon to be ADS-B systems.
It is unequivocally NOT your place to attempt to contact or otherwise interact with the distressed aircraft. This will only increase confusion and create a situation where you are potentially contradicting what ATC is saying. To address your specific situation, controllers on the ground have the ability to get the required people on the line to work the problem. If someone with suicidal intentions is on the frequency controllers will respond (and in this case did respond) in a proper manner. It is not your place to attempt to talk to them as you are likely unqualified to do so and will likely worsen matters.
As a slight caveat to the above statement, there is one instance in which you as a pilot should get involved. In some areas of the country (or the world for that matter) where facilities are far apart and radio coverage at low altitudes is limited you may be in a situation where you can relay important information to a station that is with in your range but out of range of the aircraft in distress. In this case it is important to know this is the case and that you are not simply in range of an aircraft but out of range of the facility they are communicating with. If the distressed aircraft is making repeated, uniform calls with no clear response you may consider contacting approach and asking if they are aware of the situation. This is a very specific case and should be handled as a one-off on a case-by-case basis.