Was it Alice fault that Edward believed Bella had died?

No it is not Alice's fault. It's no one's actually.

Reason why is that Alice only caught a glimpse of Bella jumping off the cliff. Alice can't see Bella's future whenever she is around the Werwölfe and Bella went cliff diving without anyone around. When Alice saw that vision, she rushed to Bella's house. She didn't know what happened; she really did think that Bella was trying to hurt herself.

And, of course, when Alice saw that vision, she told her family. Alice really did expect Bella to have passed away when she went to Bella's house. When Rosalie called Edward, she wasn't aware that it was sort of a misunderstanding and I certainly think that she didn't know Edward would rush to the Volturi.

In addition to what mandapanda said, Alice only knew for a few Minuten before Rosalie told Edward
Her and Bella talked for about a minute, she chastised her for a bit, then got distracted Von the wolf smell. Bella told her about Jacob, then she chastised Bella a little more, then Jacob walked in and they had a little stare down, then Alice stepped out. Edward called the house and Jacob answered.
Right after that, Alice had the vision.

There really wasn't much opportunity for Alice to call anyone.

yes

No it wasnot Alice's fault she honestly believed that Bella commited suicide, because she was herz broken about Edward.

No of course not.Alice's visions are subjective and there are certain blind spots to her visions.She didnt know about Jacob being a wolf and didnt see him pull Bella out of the water.As for Alice not telling the rest of the Cullens Bella was alive,they were in different places after they left Forks,so there was no way she could.But if Du recall she did get in touch with Rosalie after seeing Bella was alive,so Rosalie must have told the rest of the Cullens Bella was alive.