When Sen. Joseph R. Biden Jr. (D-Del.) recently addressed the NAACP convention, he took on the Rev. Jesse L. Jackson -- sort of -- on his own turf. Jackson plans to do the same to Biden, except it won't be sort of. Jackson travels to Delaware Sunday for a series of events "to demand a present part of the dialogue . . . to make certain the Democrats and the people in Delaware are aware of all sides of the debate that is emerging about the direction of the party."

Biden, a potential 1988 presidential candidate, told the convention to "reject the voices . . . who tell you . . . that only blacks should represent blacks." He did not mention Jackson, another potential for '88, by name. Only later, talking with reporters, did he name Jackson and expand on his criticism.

At lunch yesterday at The Washington Post, Jackson said he had thought Biden was going to make "a bold and courageous move," in addressing the NAACP, "because I thought I was going to watch him convince the NAACP convention why his antibusing and tuition tax credit and Gramm-Rudman-Hollings positions were in their best interests."

But instead, Jackson said, Biden "preached away from the storm and then after the people had left he talked with the press about what was left out of the speech." This, Jackson said, "is not the way to get into the second volume of 'Profiles in Courage.' " Hart Adds an Aide

With six months remaining in the Senate, Gary Hart (D-Colo.) is bringing on a new special assistant: Dick Murphy, former legislative director of the Service Employees Union. Murphy will "advise Hart on a wide range of economic issues," said Hart aide William Dixon.

Murphy also brings Hart access to the labor union community. Although Hart's voting record gets a good rating from labor, the 1984 presidential campaign left behind some bad feelings because Hart was critical of labor's role in Walter F. Mondale's campaign. Since the election, Murphy said he has tried to "smooth things over" between Hart and labor.

As for the future, Murphy said it is "reasonable to conclude that I'm interested in helping him in his presidential campaign, if he declares." Cuomo Hits the Road

New York Gov. Mario M. Cuomo (D) will make one of his rare fund-raising trips outside his state when he visits Washington July 30 to raise money for Rep. Stan Lundine, his choice for lieutenant governor. Lundine faces a Sept. 9 primary challenge from two very rich candidates, real estate developers Gilbert Di Lucia and Abraham Hirschfeld. Hirschfeld has already spent $1 million and has said he will spend at least $3 million. AIDS Agreement

Sen. Alan Cranston (D-Calif.) and his Republican challenger, Rep. Edwin V.W. Zschau, agree on at least one issue facing the voters: They oppose an anti-AIDS initiative put on the California ballot by supporters of Lyndon H. LaRouche Jr. The measure could lead to quarantine of AIDS victims and a ban on their working in restaurants and schools. Cranston and Zschau say that the initiative's threat of isolation could discourage those at high risk of AIDS from seeking a diagnosis and/or treatment.

Both have signed an opposition argument to be printed in ballot information pamphlets to be mailed to voters shortly before the election. Running From Office?

In Massachussetts, only 501 candidates are officially running for office, about half the traditional number and "the fewest number of candidates in the state's history," John Cloonan, director of state elections, told The Boston Globe. He attributes the drop to new financial-disclosure rules and to the insecurity and low income of public office.

But in Oklahoma, running for office apparently has not lost its luster. During the three-day filing period last week for the Aug. 26 primary, 13 candidates filed for governor, nine for lieutenant governor, eight for treasurer and five for attorney general. A total of 673 candidates filed for 322 offices, according to the State Election Board.