"Could it be that the lithium battery has completely died and I should replace it?"   Indeed, try a new battery first...they're not that expensive and it might fix the problem.   
Hey so for some context; The 3 year warranty expired on December 27, 2016. Last week my alienware started to go through a process where it would boot up for about 30 seconds and shut off again without ever feeding any signal into my monitor. I did some searching and found that by resetting the CMOS it could potentially fix the problem, and so that I did and voila, it fixed the issue.Anyways, come today the same issue came back, the computer was turning on for 30 seconds at most and shutting off again without providing any signal to the monitor. I tried to reset the CMOS by moving the pin again from 0 [0 0] to [0 0] 0 and back again. However, this time the computer has completely died on me. I have ensured the CMOS is in the right spot and when I test the power supply unit the green light comes on and I can see that power is coming into the computer because the fans start along with the graphics card and alien head push start button.Could I have fried the CMOS? There is no physical evidence of that but I am not sure. Could it be that the lithium battery has completely died and I should replace it? Or could it be something deeper?
Area-51 R2, dead/won't turn on