The torah relates how both Yosef HaTzadik and Moshe Rabbeinu married a bat kohen -although in both instances their father’s where kohen’s to עבודה זרה.
Rabbeinu Bachya, quoting Rabbi Dan, explains that Moshe chose to marry a daughter of a kohen as the kohen was exempt from all royal taxation, thus making it less likely that pharoh come after him for his killing of the Egyptian officer. Nontheless, we glean from these stories that the בת כהן was deemed a person of importance. The torah goes on to place the bat kohen in a very special place, she is privileged to consume kodoshim kalim (minor kodoshim) as is written; כל תרומת הקדשים ..נתתי לך ולבנתיך אתך
Specifically, the bat kohen may consume of the four breads that accompany the todah offering, from the two breads that are brought along with the ram of the nazir, and from the cooked foreleg of the korban nazir, she only may consume trumah, but may also give trumah to her servants (mishna torah). The school of Rabbi Eliezer ben Yaakov permits giving her the foreleg, cheeks and abomasum. Reishit haGez may be given to her, and she may eat of the bchor animal. Midioraita (basic torah law) the bat kohen is given trumah, trumat maasar and challah -even in the absence of her kohen father, but midirabbanan -rabbinicaly- these three aren’t given to her for reason that she may be divorced and thus forbidden to her.
Marriage to a kohen
Since it is established; a kohen, all the while that he is able to marry a bat kohen shouldn’t marry a bat yisroel, as is said in psachim; (the) “optional” seudah (i.e. not a seudat mitzvah, is one where) a bat yisroel(‘s marriage ceremony) to a kohen
Generally, the bat kohen is considered an excellent “shidduch“. Rabbi Menachem Meiri describes the average kohen-father as one who is dedicated to torah study and torah instruction, thus influencing the bat kohen to a lifestyle of torah dedication;
It is thus said “he who wishes to become wealthy should cleave to the progeny of Ahron” as regarding the kohen the torah says “they will instruct your laws to (the children of) Yaakov, and your torah to the children of Israel”. 
The Shulchan aruch harav interprets the notion that the bat kohen should wed a kohen quite seriously by stating that one must backtrack to burn his chometz if en route to the wedding ceremony of a bat kohen to a non-kohen -as that ceremony isn’t deemed a seudat mitzvah.
The Mishna writes how the bet din shel kohanim would require the amount of 400 zuz for the marriage of the bat kohen. -a two-fold increase of the standard 200 zuz amount. This entitlement is discussed in talmud bavli and endorsed by the rishonim (Nimueki Yosef, R”Y of Luniel, Rabbeinu Asher) although no record of this increased ktubah has been recorded since the times of the amoraim.
A different understanding of the mishna is given by talmud yerushalmi; the exaggerated amount was enacted by the bet din shel kohanim as a penalty. The penalty being in both instances, one where the bat kohen married a non kohen and in the instance where the bat yisroel who married a kohen, the amount of 400 zuz was listed in their ktubah amount. Reason being as chazal wanted each to cleave to their kind and the exaggerated ktubah would serve as a deterrence for intermarriage.
The torah expects tzniut behavior from the bat kohen, the torah rules that the bat kohen who engages in harlotry is punished by sreifah -a more serious punishment than the standard skillah- (mishna san. 9:1, sanhedrin 66b)
- Rabbeinu Bchaya to Shmoth 2:21^
- Meiri to Yevamoth p. 314^
- Sefer HaChinuch, Mitzvah 267^
- pesachim, yalkut shimoni; ”ובת איש כהן כי תהיה לאיש זר” וסמיך ליה “ובת איש כהן כי תהיה אלמנה וגרושה” א”ר יוחנן בת כהן לישראל אין זיווגן עולה יפה מאי היא? א”ר חסדא או אלמנה או גרושה או זרע אין לה, במתניתא תנא או קברתו או מביאתו לידי עניות. איני, והאמר רבי יוחנן הרוצה להתעשר ידבק בזרע של אהרן כל שכן שתורה וכהונה מעשרתו, לא קשיא הא בת”ח הא בע”ה. -yalkut shimoni to vaykra, p. 738 of ha’maor edition^
- Bet HaBchirah to psachim 49a^
- The girsa of the Tosefta is “bet din” instituted such (without mention of bet din shel kohanim). Yisroel families that had documented lineage also would write the amount of 400 zuz, but in that case it was written as a bonus, whereas by the ktubah of a bat kohen is was written as the base amount -Rabbeinu Tam quoted by the mordechai to ktuboth minor ch. 136^
- מצב גביית ד’ מאות היה בתוקפתה בימי תחילת תקופת האמוראים ומאז ואילך שוב לא מצאנו -תולדות הכתובה בישראל ד’ מח , א. אפשטיין ^