To reach Kawasaki from Narita Airport, take the JR Narita Express to Tokyo or Shinagawa and transfer to the Tokaido Line. This takes about 90 minutes and the trip is covered by the Japan Rail Pass and JR East Rail Pass. Without a pass the one-way fare is ¥3280; for only ¥220 more you can travel to Kawasaki using the Suica & N'EX Combination Ticket. (See the JR Line section of the Narita article for more details)
A cheaper way of traveling from Narita Airport is by using a combination of commuter trains on the Keisei Railway, Toei Asakusa Subway Line and Keikyu Line. This takes about 2 hours with at least one change of trains required en route, and the trains can get crowded, but the fare is only ¥1380.
From Haneda Airport, you can take the Keikyu Line's Airport Express to the Keikyu-Kawasaki station in 15-20 minutes for ¥400. Note that the Airport Express has several variants: If the train's destination is Shin-Zushi or Kanazawa-Bunko then you can stay on the train for the entire trip. If the train goes to another destination, it's likely to continue on into Tokyo so you will need to change at Keikyu-Kamata station to the next main line train on Platform 2.
JR Kawasaki station is on the Tokaido Main Line from Tokyo. Trains on both the Tokaido Line and the Keihin-Tohoku Line stop here. The Tokaido Line is slightly faster from Tokyo Station (15 minutes) compared to the Keihin-Tohoku Line (22 minutes); it costs ¥290 in either case. From Yokohama it takes 7 minutes on the Tokaido Line and 13 on the Keihin-Tohoku Line (¥210).
The Nambu Line runs from Kawasaki along the western part of Tokyo. It runs to Tachikawa, a stop on the JR Chuo Line (55 minutes, ¥620), and also offers connections to the Tokyu Line at Musashi-Kosugi, the Odakyu Railway at Noborito, and the Keio Railway at Bubaigawara.
The immediately adjacent Keikyu-Kawasaki station is also accessible more cheaply on the private