Question:How should a Christian deal with a person who is easily angered? Well, let me rephrase that -- how can a Christian deal with a very easily angered father, who is always speaking negatively about people, things, and me?
I'm not certain how old you are, so let me cover a few points. I'm not saying that these are true in your case, I'm just making sure I am giving you a complete answer.
There is a phase during adolescence where young people, male and female, are unable to read the body language of others. I have papers in my files on studies that have been done. Teenagers tend to rate expressions and body language as being far more severe than what is actually being portrayed. Thus, a parent's look of annoyance is taken as the parent being angry with the teenager. I was with a young man and his family at a restaurant several years ago. The young man's father was getting really annoyed at the young man because he wouldn't tell the waiter his order. Since he was sitting next to me, I asked him what was going on. He whispered that he thought the waiter was really mad at him. I asked him how he came to that conclusion and he said the way he looked at him. The waiter was a bit gruff, but what I saw was a man having a rough day and who was trying to be pleasant even though he didn't feel like it. I reminded the young man about talking about misreading body language and then pointed out several other possibilities for the man's behavior. By the end of the meal the young man decided that the waiter wasn't so bad after all (the waiter managed to smile at him while serving the platters).
I said all of that to say that if you are in your teenage years, be aware that many things may not be as extreme as you perceive them to be.
Next, I want you to think about why your father is so angry. When a person is generically angry at the world, as you describe your father, there is usually more going on that the immediate things he is blowing up at. Anger is one of those emotions that accumulate. Things can annoy you, but get too many annoyances and your emotions overflow your self-control. So ask yourself if your dad is under a lot of stress from something else and, thus, has a short fuse. Related to this is some people in the face of stress try to medicate themselves with alcohol or drugs. While under the influence, problems don't seem as bad or are easily ignored. But the problems don't go away and the emotional response to those problems have been pent up behind a dam. As the person comes down, the dam breaks and all the pent up emotions come out at once.
Understanding won't stop the problem, but it might make you aware of the times when rage is more likely to appear. Then you can avoid triggering an unwanted response. "If the ruler's temper rises against you, do not abandon your position, because composure allays great offenses" (Ecclesiastes 10:4). So here is the first answer: you have to learn how to keep control of your own emotional responses. Now, if you are a teenager, that is easier said than done since your own emotions are already bouncing all over the place. Yet, even though it is hard, the effort will make you a better person. What too many people do is respond to anger with anger. The other person then increases their anger in order to "win." Which then leads to an even great response of anger. And there is no stopping point. "A gentle answer turns away wrath, But a harsh word stirs up anger" (Proverbs 15:1). Eventually people do things they later regret. "An angry man stirs up strife, And a hot-tempered man abounds in transgression" (Proverbs 29:22).
In this, you need to learn to control your emotional responses. "This you know, my beloved brethren. But everyone must be quick to hear, slow to speak and slow to anger; for the anger of man does not achieve the righteousness of God" (James 1:19-20). You can't control your father's emotions, but you can put a reign on your own. Work on what you can control and let God handle what you can't.
Finally, you know that your father speaks negatively about everything. So is he a good judge of who you are? I hope you realize the answer has to be "no" because a biased judgment is never accurate. A person who always tells you how bad you are and a person who never sees a problem with you are both equally worthless. The problem is that we don't mind being told we are good when we know we really weren't, but we have a huge problem being told we are bad when we know we were right. Don't give credence to flawed judgments. Another person's opinion doesn't change who you are. When you get advice from someone you respect, it might hurt, but think about it and make changes in your life. "It is better to listen to the rebuke of a wise man than for one to listen to the song of fools" (Ecclesiastes 7:5). So let your father's hateful, unthinking words, just roll off you.