Note: To follow is a mere synopsis of the subject. In all cases of question or doubt one is to bring the issue to a Beis Din that is familiar in these matters.
If one says he is Jewish because his mother is Jewish is he to be believed?
In general, if a person says he is Jewish and acts like a Jew then he is believed to be Jewish. Some Poskim rule that this applies even if the person came from a different country. Other Poskim however rule that if he came from a different country then he must prove his Jewishness, even if he acts like a Jew. Practically, we no longer rely on the above Chazaka regarding Jews who come from areas where intermarriage is common, such as the USSR. This especially applies if there is a reason to place doubt into the persons Jewishness.
If one says he is Jewish because he converted, is he to be believed?
If the person was known as a gentile and now claims to have converted, then he is not believed to be allowed to marry a Jew unless he supplies witnesses or re-converts. This applies even if he acts like a Jew. If however the person was held to be Jewish and says that in truth he converted, then he is believed without witnesses. Now although some Poskim argue that even such a person must bring witnesses nevertheless, the custom is like the Michaber.
How to verify the Jewishness of a questionable Jew:
The verification of one’s Jewishness can be done in several ways. Some Poskim suggest the following criteria:
- Checking the legal identification of the person or his mother, and seeing if it says that he or she is Jewish.
- Verifying the Jewish names of the person and his family.
- Knowledge of Yiddish, and acting like a Jew or knowledge of Jewish customs.
The person in question must fulfill at least two of the above three conditions to be considered a Jew, and only in a case that there is no other reason to cast doubt on his Jewishness. In today’s times, one also heavily relies on the opinion of an investigator who is trained and experienced in verifying whether the person’s claim is true or not. [So is done today in all Batei Dinim].
 Michaber Even Haezer 2/2; Rashal Kesubos 2/40; Rabbeinu Tam in Yevamos 46b; Rashba; Ramban; Pischeiy Teshuvah 2/2; Aruch Hashulchan 2/12
The reason: As people contain a Chezkas Kashrus, and majority of people who say they are Jewish are Jewish. [ibid]
Other opinions: Some Poskim rule one is not believed to say he is Jewsih unless he brings proof. [Rameh; Rashi, brought in Tur 2; Beis Shmuel 2/2 suspects for this opinion]
 Shach 268/21; Bach Yoreh Deah 268; Halef Lecha Shlomo Even Haezer 15
 Beir Heiytiv Even Haezer 2/4 in name of Maharit 1/149 and Beis Hillel, and so rules Michaber Y.D. 268/10 regarding a gentile who says he converted; See however Aruch Hashulchan 2/13 that this only applies to an individual, while a family does not have to prove their Jewishness if they act like a Jew, as a family maintains a Chazaka.
 Nevertheless, regarding Tefillin, seemingly one may rely on the Chazaka if the person acts like a Jew and says he is Jewish, and so is the custom. [See Michaber 268/10 that we are only stringent to require proof in order to marry the person]
 Michaber Y.D. 268/10; Yevamos 46b
 Michaber ibid
 Rambam in Michaber ibid
 Shach 268/21 in name of Bach
 Opinion of Rav Moshe Feinstein
 Opinion of Rav Moshe Feinstein
 Nevertheless, the above applies regarding allowing the person in question to marry a Jew. However regarding Tefillin, seemingly one may rely on a more lenient stance, so long the person acts like a Jew and says he is Jewish. [See Michaber 268/10 that we are only stringent to require proof in order to marry the person]