For the franchise history, that is 31 games by Vladimir Guerrero in the 1999 season. For the Washington Nationals part of the franchise history, that is 30 games by Ryan Zimmerman in the 2009 season.
For the 2009 season, through games played on May 2 that is 21 games by Ryan Zimmerman of the Washington Nationals.
Pete Rose had a 44 game hitting streak in 1978.
George Brett had a 30-game hitting streak.
30 by Willy Taveras in 2006
35 by Luis Castillo in 2002
31 by Vladimir Guerrero in 1999
28 by Gabe Kapler in 2000
39 by Paul Molitor in 1987