The pronoun 自己 is used in the following meanings:
1) "one's own (oneself)"
It is placed before a predicate and describes the way of action. It can be used independently or after other personal pronouns. For example:
You have to cook by yourself
nǐ bìxū zìjǐ zuò fàn
I'll say it myself, thank you!
wǒ zìjǐ huì shuō, xièxie le
In this case, 自己 serves as an object and means "to do in relation to someone".
Don't let yourself get into bad habits
bié ràng zìjǐ rǎn shàng èxí
Did you hurt yourself?
nǐ shāngzháo zìjǐ le ma
3) "own" (possession).
In this case, 自己 serves as an attribute and is used with the particle 的 after it.
It was her own idea
nà shì tā zìjǐ de zhǔyì
They don't want to express their feelings
tāmen bú yuàn biǎodá zìjǐ de gǎnqíng
The adverb 亲自
The adverb 亲自 placed before a predicate.
In this case, 亲自 can be replaced with 自己.
I cook by myself every day
wǒ měitiān dōu huì qīnzì zuòfàn
亲自 is most often used in formal language when referring to a high-status person. For example: a boss, a president or a minister.
The president has personally taken up this question
zǒngtǒng qīnzì chūmiàn chǔlǐ zhè ge wèntí
Presley himself sings in this recording
zài zhège lùyīn zhōng pǔléisīlì qīnzì chàng
If a person has personally done something, it is especially valuable. For example, it may be a rare or unusual situation.
In the last two cases, 亲自 cannot be replaced with 自己.
He wants to thank his rescuers in person
tā yào qīnzì qù gǎnxiè yíngjiù zhě
He trained these people himself
tā qīnzì péixùn zhèxiē rén
自己 can serve as an adjective "own" or a pronoun "oneself", whereas 亲自 cannot. 亲自 has only one meaning — personally, in person.