[adjective] + 死了
[verb] + 死 + [person] + 了
死了 after an adjective pushes it to the extreme, like "to death" or "so" in English. It is casual speech, and it mostly goes with feelings you'd complain about: tired, hungry, hot, annoyed.
With verbs of feeling, the person affected can go between 死 and 了. 气死我了 is "it's driving me mad".
A few positive words take it in very casual speech, mainly 高兴 and 开心.
死了 already says "extremely", so it takes no 很 or 非常 in front.
It is too casual for writing or for a polite complaint at work. There, use 非常 or 特别.