Can giving money to my parents and relatives be considered as tzedakah?
If your parents are in need then give them because Tzedaka is a Mitzva to everyone, but even if they dont need so much helping them is still a mitzva see Isiah 58,7:
הֲלוֹא פָרֹס לָרָעֵב לַחְמֶךָ, וַעֲנִיִּים מְרוּדִים תָּבִיא בָיִת, כִּי תִרְאֶה עָרֹם וְכִסִּיתוֹ* *וּמִבְּשָׂרְךָ לֹא תִתְעַלָּם Give to the hungry your bread, bring the poor to your house when you see the unclad cover them and from your flesh don't turn away
The Gemara Kesubos 86a explains from וּמִבְּשָׂרְךָ לֹא תִתְעַלָּם that there is specifically a Mitzva to help a relative even when they are not in need of Tzedakah rather just help that has a consequence of monetary value.
Asking a general question requires a general and detailed answer:
The idea of Tzedakah in the Torah is very broad and applies to practically all good deeds that are just (צדק) in the eyes of G-d and people - from learning Torah to keeping all the Mitzvot to. (see צדקה - ביאור for the list of Torah verses)
Traditionally, the Sages narrowed the use of this word down to the Mitzvah of helping the needy only (ibid). And so it appears in all Halachic books, see Rambam (Matnot-Aniyim)
The Sages did not define a needy very clearly, in terms of his income or lifestyle and whether the need is truly physiological (nothing to eat) or psychological (wants a better life).
From this, we can infer that giving money to any [Jew] can be considered a Mitzvah of Tzedakah as long as one is not obligated to pay: either as a debt (paying a loan or a salary) or a Halachic obligation (a fine, woman's Ketubah, supporting kids under age of 6 etc.).
To break down your question:
Regarding the parents, the Halacha concludes that one is not obligated to spend his own money on honoring the parents, therefore supporting them falls rightfully under Tzedakah.
Supporting the kids is a big Machloket, how much, for how long and how strong is one obligated to support them, so giving $5 to 5yo boy to buy lunch at school is probably NOT Tzedakah as he's obligated to feed him, but buying a $950 iPhone to a "needy" teenager might be.
Supporting the wife is an obligation and cannot fall under Tzedakah. So if you buy your wife a diamond ring, don't call yourself a Tzaddik!
Supporting in-laws and grandparents and more distant relatives is a true Tzedakah, as there's no obligation to do so.
The Mitzvah of Tzedakah is unrelated to one's intentions, so even if one does it to brag or expect a favor, it still counts as Tzedakah.