Kennedy died last week of brain cancer at age 77.
Coakley said at a news conference the state has had a "crisis of confidence" following Kennedy's death and she wants to pick up his mantle.
"We've depended on him here in the Commonwealth and in Washington, and we will miss his strength and leadership and his sense of humor. As some have noted, no one can fill his shoes, but we must strive to follow in his footsteps," she told supporters at a downtown Boston hotel.
Coakley sidestepped a question from reporters whether she favored the changing state law to allow the governor to appoint an interim senator, as Kennedy had requested in a letter before his death. Legislators hold a hearing on the matter next week.
"For me, personally, I am fully focused on the race," she said, adding she trusted legislators "will make the right decision."
The 56-year-old Coakley becomes the most prominent candidate to officially declare. Several others are waiting for Kennedy's nephew, former Rep. Joseph P. Kennedy II, to decide if he will run.
Coakley said in reponse to a question she decided to commit to the race without waiting to see if a Kennedy family member would run because of the senator's own declaration in his letter seeking an interim appointment that the state needs strong and effective representation in Washington.
She said she decided to run "because government should work well and work for everyone," adding that the performance of government "seems at times disheartening and discouraging."
"I believe now is the time to move beyond the idea of `good enough' government and demand a new standard of excellence. I know that I need to prove to voters that I am the best candidate in this race and I believe I can do that," Coakley said.
Coakley described her humble roots in Western Massachusetts and her career as a prosecutor, both as Middlesex District Attorney in a large district that includes populous suburbs of Boston, and since, 2007, as the state's attorney general.
"Now I hope to bring my experience to Washington," she said.
She did not overtly highlight her status as the state's top female elected official.
Other potential Democratic candidates include U.S. Reps. Michael Capuano and Stephen Lynch. Potential Republicans include former Lt. Gov. Kerry Healey and state Sen. Scott Brown.
Former Boston Red Sox pitcher Curt Schilling is considering an independent campaign.