I met this person yesterday.

There was an unknown person in the meeting.

A person means a living human.

There is someone I want you to meet.

Someone is knocking the door.

I am lost with sentences of the above type. biri, kişi, şahıs, kimse? Could a native explain the differences (or translate the sentences with all the possible alternatives)?

I would say:

I met this person yesterday - Dün bu kişiyle/şahısla tanıştım. (I think we can´t use "biri" when we are talking about a specific person. For example we can´t say "Dün bu biriyle tanıştım." but if your sentence was "I met a person yesterday", it could be "Dün biriyle/bir kişiyle/bir şahısla tanıştım." If you say "Dün şahısla tanıştım" it means "I met the person yesterday" )

There was an unknown person in the meeting. Toplantıda bilinmeyen biri/bir şahıs/bir kişi vardı.

A person means a living human - Kişi/Şahıs/Birey yaşayan bir insan demektir. (In Turkish definition sentences, indefinite article "bir" is not used, like "Kağıt, üzerine yazı yazılan bir şeydir. | A paper is something you write on. )

There is someone I want you to meet - Tanışmanı istediğim biri/bir şahıs/bir kişi/bir kimse var. (you can use "bir kimse" in the meaning of someone but I wouldn´t prefer using that because it sounds a bit unnatural to me) 

Someone is knocking the door - Biri/Bir şahıs/bir kişi/bir kimse kapıyı çalıyor.

"Kimse" is generally used in the meaning of "hiç kimse". For example:

-Is someone knocking the door | Biri kapıyı mı çalıyor?

-No, there is nobody here | Hayır, burada kimse yok. 

Great answer, srhat, thank you.

kimse is typically negative I know. That is why its use in affirmative sentences is quite interesting.