How can I become an Israeli citizen?

You certainly qualify under the Law of Return, having one Jewish parent as you do (the rubric IIRC is having one Jewish grandparent). Ask your nearest Israeli consulate or embassy.

You should talk to the consulate, of course, but I'm pretty sure you're good to go. For immigration purposes, Israel uses the Nuremberg definition: i.e. anyone the Nazis would have persecuted for being Jewish is jewish enough to get in.

I'm not Israeli -- or Jewish -- but could your father qualify and then grandfather (or father, I guess) you in?

Contact the Israeli consulate in NYC and ask them this question

*Eligibility requirements Those who immigrate to Israel under the Law of Return are immediately entitled to citizenship. However, differences of opinion have arisen as to whether a person who claims citizenship under the Law of Return should be automatically registered as "Jewish" for census purposes. According to the halakhic definition, a person is Jewish if his or her mother is Jewish, or if he or she converts to Judaism. Charedi citizens of Israel generally do not recognize conversions performed by Reform or Conservative Judaism. However, the Law provides that any Jew regardless of affiliation may migrate to Israel and claim citizenship. Originally, the Law of Return was restricted to Jews only. A 1970 amendment, however, stated that, "The rights of a Jew under this Law and the rights of an oleh under the Nationality Law... are also vested in a child and a grandchild of a Jew, the spouse of a Jew, the spouse of a child of a Jew and the spouse of a grandchild of a Jew".* Your mother is not Jewish, you did not convert, hence you aren't Jewish according to Jewish Law. If you would want to become a citizen of Israel - you'd have to become one according to the http://www.mfa.gov.il/MFA/MFAArchive/2000_2009/2001/8/Acquisition%20of%20Israeli%20Nationality

"For the purposes of this Law [of Return], "Jew" means a person who was born of a Jewish mother, or has converted to Judaism and is not a member of another religion." More info on your options http://www.mfa.gov.il/MFA/Facts+About+Israel/State/Acquisition+of+Israeli+Nationality.htm

You will have no problem becoming an Israeli citizen if your father is Jewish.

I am in the same position (have a Jewish farther and non-Jewish mother) and I have immigrated to Israel and now am an Israeli citizen. To simplify the qualifications for the right of return, you have to have at least one Jewish grandmother to be eligible, or be an immediate family member of a such person (spouse or child) and immigrate as a part of the family, or be converted into Judaism. In my case, for example, myself, my spouse, and my children immigrated together based on my eligibility and we all became Israeli citizens. So you are good to go. By the way, you do not need Israeli visa to enter Israel for purpose of immigration according to right of return, but you will have to demonstrate that you are qualified. Therefore it is advisable to check whatever documents confirming your eligibility in Israeli consulate first.

A 1970 amendment, however, stated that, "The rights of a Jew under this Law and the rights of an oleh under the Nationality Law... are also vested in a child and a grandchild of a Jew, the spouse of a Jew, the spouse of a child of a Jew and the spouse of a grandchild of a Jew". watercarrier, you're missing that the questioner is the child of a Jew--the father.