The play portrays Edward very much as a weak, lax, self-indulgent king interested only in the pursuit of pleasure with his beloved Galveston, wholly neglecting affairs of governance. Not surprisingly this provokes his subjects; power-struggles ensue, and he is deposed and eventually killed. Throughout, he appears petulant, stubborn, unreasonable, and quite unfit to be a ruling monarch.
It is true that at times he does show something of a different nature, for instance his moment of wrath when Gaveston is killed (Act 3, scene 2), and he even shows some resolution on the battlefield (Act 4, scene 5). However he can't sustain this show of spirit, and so never really achieves anything. He remains quite ineffective throughout the play.
The way Edward meets his death only serves to reinforce this sense of weakness. He is forced to endure degrading conditions in his prison, and he meets his fate sobbingly, ever protesting his own helplessness. Almost his last words are on this theme:
I am too weak and feeble to resist. (V.v.110)
This is exactly how he has appeared throughout the play; he has not been able to withstand attacks on his position, nor the blandishments of Gaveston. He cuts a very passive and helpless figure.
Edward's general lack of dignity makes him a pathetic, rather than a tragic figure. Tragic heroes show some strength of character, but Marlowe's Edward seems to have very little, or none.