How can you say "I will miss you" and "I hate you" in korean?

Just once I want to hear the story behind why these phrases are needed.

In this case I imagine you are saying "I will miss you" to your boyfriend, and "I hate you" to his wife.

Anyway, Yoshin is right on the translation. Noiro is right on the usage, if you are saying goodbye at that moment. Use what Yoshin said if you are talking about the future when the person will leave. It may sound awkward but they will get the point.
Also, 鞁柎 literally means 'don't like' and 氙胳泴 literally means 'strong hate/ugly person' so 氙胳泴 is a much better choice to get across the point of true disgust at a person.

i will miss you
氤搓碃 鞁鹅潉 瓯办槇鞖?(pogo siphul koyeyo)
(literally means I will want to see you)
-->> polite, casual speech act

**intimate, close friends speech act:
氤搓碃 鞁鹅潉 瓯办暭 (pogoshipulgoya)

** formal, polite speech act:
氤搓碃 鞁鹅潉 瓴侂媹雼?(pogo shipul komnida)

i will miss you
攴鸽Μ鞖?瓯办槇鞖?(keriul koyeyo)
-->> polite, casual speech act

**intimate, close friends speech act:
攴鸽Μ鞖?瓯办暭 (keriul goya)

** formal, polite speech act:
攴鸽Μ鞖?瓴侂媹雼?(keriul komnida)

i hate you
氙胳泴頃挫殧 (miwoheyo)
-->> polite, casual speech act

**intimate, close friends speech act:
氙胳泴頃?(miwohe)
or 氙胳泴頃滊嫟 (miwohanda)

** formal, polite speech act:
氙胳泴頃╇媹雼?(miwohamnida)

******************* in korean language, object doesn't have an important role (as subject also doesn't)
so basically, you just say the verb, and modify it to the meaning you want to communicate

in the case you want to add in I and you as the subject and object, add:
雮橂姅 雱堧ゼ (nanun neo-rul) for intimate speech act
鞝€電?(name)鞚?for polite speech act
before the sentence.

e.g: 雮橂姅 雱堧ゼ 氤搓碃 鞁鹅潉瓯办暭
(nanun neo-rul pogo shipul koyeyo)

Informal for "I will miss you":
氤搓碃鞁鹅潉瓯办暭.
-bogo ship-eul guh-ya.
But this literally means "I will want to see you" which sounds a bit awkward.Usually when people leave other people they say something like 鞛?雼る厐鞕€(jahl dah-nyuh-wah) which means "Have a good trip(and come back)".

Formal for "I will miss you":
氤搓碃 鞁鹅潉 瓯办棎鞖?
-bogo ship-eul guh-eh yoh.
It would also sound awkward here too so you can say 鞛?雼る厐鞓れ劯鞖?jahl dah-nyuh oh-seh yoh).

Informal for "I hate you":
雮?雱?氙胳泴頃?
-nahn nuhl mee-wuh-heh.
or
雮?雱り皜 鞁柎.
-nahn nee-gah shil-uh.

Formal for "I hate you":
鞝€電?雼轨嫚鞚?氙胳泴鞖?
-juh-neun danshin-ee mee-wuh yoh.
or
鞝€電?雼轨嫚鞚?鞁柎頃挫殧.
-juh neun danshin-eul shil-uh heh yoh.

There are many other ways to say these but these are the basic ones.
Hope I helped!

雱り皜 鞁柎 neka shileo
= I hate ya