“The sight of London to my exiled eyes, is as Elysium to a new-come soul” 

The reference to Elysium implies that after his exile, Gaveston sees London as a sort of Heaven. This makes sense in the context of his romantic relationship with the king. Elysium in myth was also a place for heroes, and Gaveston’s inflated sense of self may be another facet of his use of the name. 

“Farewell base stooping to the lowly peers! My knee shall bow to none but to the king.” 

Gaveston clearly looks down on the common people. He views it as a base insult to count himself among them. Given the influence he wields over the king, this is an ominous sign that the priorities of the kingdom will go ignored. 

“My father is deceased. Come Gaveston, and share the kingdom with thy dearest friend” 

The idea that a king shares a kingdom with its citizens should be heartening. It implies that the king is aware that he, too, is one of the people, not a deified symbol. But this is not what Edward means when he offers to share it with Gaveston. Edward sees the kingdom as a personal playground for himself and his friends.