I'm a religious Jew. I kind of lost faith and ate non-kosher purposely, and now I feel so bad with myself that I want to die. What should I do? Is there any type of repentance I should take upon myself so that I feel better and stop in general to eat non kosher?
(I am posting only a very short summary, read the source for the full details. In particular chapters 1 and 2.)
Chapter 1 Halacha 1
If a person transgresses any of the mitzvot of the Torah, whether a positive command or a negative command - whether willingly or inadvertently - when he repents, and returns from his sin, he must confess before God
This refers to a verbal confession.
Teshuvah atones for all sins. Even a person who was wicked his whole life and repented in his final moments will not be reminded of any aspect of his wickedness
Chapter 2 Halacha 1
[Who has reached] complete Teshuvah? A person who confronts the same situation in which he sinned when he has the potential to commit [the sin again], and, nevertheless, abstains and does not commit it because of his Teshuvah alone and not because of fear or a lack of strength.
[Note to the questioner: In your case this refers to being angry and acting as a result of that, it doesn't just refer to avoiding non-kosher food.]
What constitutes Teshuvah? That a sinner should abandon his sins and remove them from his thoughts, resolving in his heart, never to commit them again
The first and foremost step in Teshuvah is to whole heartedly resolve not to do the sin anymore, the next step is (a positive Mitzvah) to verbelly confess and ask Hashem for forgiveness. See Tanya first chapter of Igeres Hatshuvah.
(In Shaulchan Aruch Orach Chaim 334:26 it talks about Teshuvah for Chillul Shabbos.)
I'm more concerned that you want to die. Why would you want to die after eating non kosher food? You can break nearly any mitzvah to save your life, certainly kashrut laws are included in Pikuach Nefesh. So I just find it concerning that you'd want to lose your life after committing an aveira. Do teshuvah and you're fine.