Fulk was only king of Jerusalem in right of his wife, Melisende, who was the eldest child of Baldwin II, second king of Jerusalem, and who had no sons. The line of succession in Jerusalem was through her, not him, and Melisende was Fulk's second wife. Fulk's first wife, Ermengarde, had died in 1126 at the age of about 30, after having given birth to Geoffrey.

In 1127, Fulk, a powerful lord who was respected for his political and military skills, was approached by Baldwin II (who would die in 1131), who offered Melisende's hand in marriage to Fulk. By making this match, Baldwin intended to safeguard his heirs as kings of Jerusalem. Baldwin knew that, without a husband who commanded the respect of Jerusalem's fractious noblemen and neighboring Crusader states, Melisende's deposition and the end of his line was extremely likely. But initially, Baldwin wanted Fulk to be just Melisende's consort, and wanted her to be the sole ruling queen. Fulk, however, knew his value and held out for more, securing for himself joint rulership with Melisende as part of the bargain. Fulk left Anjou, abdicating as count in favor of Geoffrey, for Jerusalem in 1129, and married Melisende in that city that year.

Therefore, when Baldwin II died, Fulk became the king in fact and in name, rather than just a prince-consort or similar title. However, Baldwin II wasn't done. Just before he died, he secured for Melisende sole legal custody of her son by Fulk, Baldwin III, who was born in 1130. This helped ensure that Fulk didn't try to usurp the throne for his own family and put Geoffrey or his other son by Ermengarde (Elias, Count of Maine) on the throne of Jerusalem.

So, essentially, Fulk split his house into two branches, one in Anjou (and later England) and one in Jerusalem. However, later, after several dynastic turnovers in Jerusalem, the kings of England did claim to be Kings of Jerusalem. In fact, to this day, the kings and queens of the United Kingdom maintain this claim to Jerusalem.

Of course, the British monarchs are not the only ones who claim to be kings of Jerusalem. Due to the aforementioned dynastic turnover, there were many houses that, at one time or another, produced a king or queen of Jerusalem. As a result, most of the surviving royal houses (both still reigning and deposed) claim to be kings of Jerusalem. The title is actually actively in use among the titles of the King of Spain, though of course Spain doesn't rule any Jerusalemite territory.

I hope that actually explains things. I went back and forth on the organization of this whole mess because dang is it tangled.