D.C. City Council member John L. Ray began his campaign for the Democratic nomination for mayor yesterday, claiming that Mayor Marion Barry, his one-time political benefactor, has mishandled the city's financial problems, run a wasteful and inefficient bureaucracy and failed to stem the rising crime rate.

With 300 supporters and friends cheering him on at the New Samaritan Baptist Church on Capitol Hill, the one-term council member used the city's error-prone water billing system as a metaphor for what he claimed was the Barry administration's incompetence.

"We cannot be proud of a government that has spent three years tripping over its own feet," Ray said.

Ray, a politically hungry lawyer who briefly sought the nomination for mayor in 1978 before dropping out and supporting Barry, became the first to formally enter this year's Democratic mayoral primary, scheduled for September.

Barry is expected to announce soon his plans to run for a second four-year term. Several other prominent Democrats, including council members John A. Wilson, Betty Ann Kane and Charlene Drew Jarvis, as well as former Carter administration cabinet member Patricia R. Harris, have also expressed interest in challenging Barry.

Council Chairman Arrington Dixon, who also flirted with the idea of running for mayor, is now expected to seek reelection as chairman.

Ray said yesterday that his top priority would be the improvement of city services, including issuing more accurate water bills, speeding up the processing of building permits and improving trash pickups.

But Ray, a 38-year-old liberal Democrat, repeatedly returned to law-and-order themes throughout his formal remarks yesterday.

He insisted that Barry had permitted serious street crimes to flourish. If elected, he said, he would push hard for enactment of mandatory minimum prison terms for persons convicted of violent crimes or the illegal sale of narcotics.

"The John Ray administration will not tolerate a crime rate that goes up and up and up, and scares the living daylight out of us all," Ray said. "And we cannot tolerate the flourishing drug traffic which threatens to take over the streets where we live and the young minds we count on for the future."

With the hoopla that often attends the start of presidential or senatorial campaigns, Ray followed up his formal declaration with a motorcade and walk from the New Samaritan Church at 610 Maryland Ave. NE, where he is a member, to his spacious campaign headquarters at 13th and E streets NW nearly two miles away.

Ray's campaign strategists, including the Bailey-Deardourff & Associates political consulting firm, plan to spend more than $500,000 to win the election, about $140,000 of which has already been raised.

Musical campaign spots for Ray began appearing on 12 Washington-area radio stations yesterday and will continue for the next two weeks.

The ads feature a specially written song entitled "This Man, John Ray" with a verse that ends:

Washington, you're the place for me.

You're my hometown, my D.C.

And to lead us along the way,

This man called John Ray.

Although his practical experience in D.C. government is limited to less than one full term as an at-large member of the council, Ray insists he could outshine Barry in coping with the city's $388-million accumulated debt and whipping into shape what he sees as a flabby, unnecessarily costly bureaucracy. Ray also said he would provide new financial incentives to remain in the city for businessmen and middle-class residents, who he says are essential to the city's long-term well being.

"I understand and have the brains to deal with the city's budget," Ray said in an interview in his city hall office Friday afternoon. "We have to start running the city like the $1.6 billion corporation that it is, and not like a mom and pop grocery store."

Ray said he plans to mount a campaign that appeals to all economic and racial segments of the city's population. However, his early efforts appear to be aimed at chipping away at Barry's support among D.C. government employes, businessmen and affluent whites in Northwest Washington.

Some of the defectors from Barry's camp who took part in yesterday's campaign festivities included Nancy M. (Bitsy) Folger of Cleveland Park, a key fund-raiser and early supporter of the mayor, who joined Ray's staff; Joseph B. Carter, a vice president of Garfinckel's who was chairman of Barry's 1979 campaign; Bobby Mitchell, a former Redskins football player and supporter of Barry who is now Ray's campaign finance chairman; Norman Neverson, a former Barry operative in Ward 4, and Marie Porter, an employe of the D.C. Department of Human Services.

Burtell M. Jefferson, who was apppointed police chief by former mayor Walter E. Washington and served briefly under Barry before retiring, is also working with Ray.

Ozzie Clay, a real estate investor and former Redskin, also was on hand yesterday for Ray's announcement, although he insisted he hasn't decided yet whom to support for mayor.

Clay said he is part of what he described as an "elite" group of businessmen that plans to raise $250,000 to try to influence the outcome of this year's races for mayor and council.

Ray, who was born in the tiny turpentine town of Tom Creek, Ga., near the Florida panhandle, moved to Washington in 1967 to attend college. He ran a long-shot campaign for mayor in 1978 as a virtual unknown and dropped out when he failed to raise sufficient funds.

He threw his support to Barry, then an at-large council member and former president of the D.C. Board of Education, who was locked into a bruising, three-way struggle with Walter E. Washington, the incumbent, and former council chairman Sterling Tucker.

After Barry won election as mayor, he successfully lobbied the D.C. Democratic State Committee to name Ray as Barry's interim replacement on the council.

Ray then won a special election in May 1979 to complete Barry's unexpired term--again with the help of Barry and his political operatives and fund-raisers. Ray was reelected in his own right in 1980 to a four-year term. If he loses the race for mayor, he will retain his seat on the council.

Barry insists that he and Ray had an "unspoken" understanding that Ray wouldn't challenge Barry in 1982--an understanding that Ray broke last August when he announced the formation of a mayoral exploratory committee.

"John understood that," Barry told a reporter several months ago. "Some things you don't have to say."

Ray denies there ever was an understanding. Somewhat defensively, he notes that in throwing his support to Barry 1979, he made it clear in a statement that Barry would have to prove himself as a chief executive if he hoped to keep Ray's support.

But Ray is sensitive to the suggestion that he broke a promise to satisfy his own political ambition.

"While I'm ambitious, I'm not possessed with blind ambition," Ray said in the interview last week. "I didn't ask anything of Marion and he didn't ask anything of me other than my endorsement."

Ray's legislative record since joining the council has been relatively modest. His lists as his major achievement working with council member Wilson to rewrite the city's condominium conversion law in a way that gives more protection to tenants. Ray contends he has been more useful to the city in helping to improve existing laws than in proposing new laws that may not be needed.

Moreover, he said that his experience and education would give him an edge in running the city.

Ray obtained degrees in political science and law from George Washington University. While still in law school, he worked for former Supreme Court Justice Abe Fortas.

After law school, Ray served as a clerk for U.S. Appeals Court Judge Spottswood W. Robinson III. Later, he worked as a counsel to the U.S. Senate antitrust and monopoly subcommittee, and then as a lawyer in the Justice Department's Office of Legal Counsel.

Ray and his wife Sarah live in a semiattached house in the Michigan Park section of upper Northeast.