The D. C. school board will swear in three new members today and is scheduled to choose a new president, with one of its newcomers--the Rev. David H. Eaton, the top vote-getter last November--expected to make a strong bid to oust incumbent board President Eugene Kinlow.

Behind-the-scenes politicking for the presidency has mounted in recent weeks, according to board members, with both Kinlow and Eaton lobbying for support. In some cases, members said, the two candidates seeking leadership of the 11-member board have offered committee chairmanships or other influential posts in exchange for votes.

Traditionally, the presidency has been a highly visible and powerful post, with the president acting as the school system's link to city government, Capitol Hill and the community. The president also controls appointments to the board's committees.

Both Kinlow and Eaton, as well as other board members interviewed yesterday, stressed the importance of harmony within the board. The board has been strongly criticized in the past for its internal bickering, although some measures of the schools' performance, such as standardized test scores in some grades, have shown improvement.

Kinlow, who has served one year as president of the board, said yesterday that Eaton apparently has a six-vote majority locked up. "It looks like Dave Eaton has enough votes," Kinlow said, "He may be the only one running, because I won't be running if I don't have enough votes. It just creates divisions."

Eaton, senior minister of All Souls Church at 16th and Harvard streets NW, was approached by some board members who asked him to run, several members said. Eaton, an at-large board member, yesterday declined comment, except to say, "I will do whatever the board feels is best for the board."

Today's election comes at a time when the board is wrestling with both new and old challenges. The system, which this year will spend more than $250 million to educate about 95,000 students, is currently negotiating salaries with four unions covering teachers, administrators, clerks, custodians and other personnel. It is grappling with the continuing problems of declining enrollment, budget woes and achievement test scores of high school students that fall 250 points below the national average. It is trying to expand a new program of annual "skills assessment" that will determine whether students are promoted. It is considering new ways to evaluate the performance of teachers in determining whether they should be retained and given pay raises. School closings also loom as a large and potentially divisive issue.

The board also is attempting to change an image marred by frequent bickering. The board has been criticized for appearing to focus more on personality clashes than on education, and it was blamed for driving a popular school superintedent, Vincent E. Reed, out of office in 1980.

Fierce battling for the presidency in recent years has produced some marathon elections, with as many as 13 ballots needed to select a winner. Several board members said they are hopeful that this year's balloting will be quicker, so that the new board can start on a unified note.

Kinlow's supporters have focused on Eaton's lack of board experience, board members said. But Eaton's supporters point to his administrative experience as a dean at Federal City College and Howard University, and to his skills in dealing with people in his years of civil rights and church experience. Eaton also was the leading vote-getter in last November's balloting.

Support for Kinlow, 41, appears strongest among the more veteran board members. R. Calvin Lockridge (Ward 8), a five-year board veteran and former president, said yesterday he supports Kinlow because he brought "stability" and because electing a new member as president "would just put us in a bind."

Kinlow and D.C. Mayor Marion Barry, formerly political allies, have clashed with increasing frequency lately over what Kinlow has seen as Barry's flagging support for the schools, sources said. These clashes, according to these sources, have weakened Kinlow's support among several of the board's newer members, who are also Barry allies.

Wanda Washburn, the newly elected Ward 3 board member, said yesterday that both Kinlow and Eaton supporters have told her she represents the deciding vote on a board that is otherwise split 5-5. She said both sides have offered her committee chairmanships. "I told both guys I am not seeking a damn thing," said Washburn, adding that she has not yet decided how to vote.

The three new board members, Eaton, Washburn, and R. David Hall (Ward 2) will be sworn in by a collection of city judges and dignitaries at 1 p.m. at Shaw Junior High School, along with the two incumbents reelected last November, Lockridge and Barbara Lett Simmons (At-large). After a reception, the election of the new president is scheduled at 2 p.m.

Electing a newcomer as president is rare for the D.C. board, which last did so when it elected Virginia Morris in 1974.