Imagine a person with no feelings, no friends, isolated both emotionally and physically. They don't feel threatened by "normal" things, making the "normal" person threatened by him. In the book, The Stranger, by Albert Camus, we embark on a journey through the mind of an existentialist. So we meet Mersault, the main character in the book, and in my opinion, the epitome of existentialism.
Throughout reading, we have observed Mersault's self-aware behavior in many scenes of the book. One instance was at the funeral.
"Some of the women were crying. She was in the second row, hidden behind one of her companions, and I couldn't see her very well. She was crying softly, steadily in little sobs. I thought she'd never stop. The others seemed not to hear her. They sat there hunched up, gloomy and silent. They would look at the casket, or their canes or whatever else, but that was all they would look at. The woman kept on crying. It surprised me, because I didn't know who she was, I wished I didn't have to listen to her anymore. But I didn't dare say anything."

Mersault doesn't seem very sympathetic to the elderly woman's feelings.

"After a while he explained, without looking at me, ?She was very close to your mother. She says your mother was her only friend and now she hasn't got anyone.' We just sat there for awhile. The woman's sighs and sobs were quieting down. She sniffled a lot. Then finally she shut up. I didn't feel drowsy anymore, but I was tired and my back was hurting me. Now it was all these people not making a sound that was getting on my nerves. Except that every now and then I'd hear a strange noise and I couldn't figure out what it was. Finally I realized that some of the old people were sucking at their cheeks and making these weird smacking noises. They were so lost in their thoughts that they weren't even aware of it. I even had the impression that the dead woman lying in front of them didn't mean anything. But I think now that that was a false impression."

In that last paragraph from the text, we see Mersault referring to his mother as the "dead woman", and not as his mother. He seems to view her as a stranger. He is unemotional but he notices almost everything about his surroundings.
Another instance in which we see Mersault exhibiting self-aware behavior was while he was watching the pallbearer and the perspiring people.
"He was walking with great dignity, without a single wasted motion. A
few beads of sweat were forming on his forehead, but he didn't wipe them off."

Another place where we see mersault being self-aware was on his balcony.
"My room looks over the main street in the neighborhood. It was a beautiful afternoon. Yet the pavement was wet and slippery, and what few people there were, were in a hurry. First, there were families out for a walk: two little boys in sailor suits, with trousers below the knees, looking a little cramped in their stiff clothes, and a little girl with a big pink bow and black patent-leather shoes. Behind them, an enormous mother, in a brown silk dress, and the father, a rather frail little man I know by sight. He had on a straw hat and a bow tie and was carrying a walking stick. Seeing him with his wife, I can understand why people in the neighborhood said he was distinguished. A little while later the local boys went by, hair greased back, red ties, tight-fitting jackets, with embroidered pocket hankerchiefs and square toed shoes. I thought they must be heading to the movies in town. That was why they were leaving so early and hurrying toward the streetcar, laughing loudly."

Another aspect of being an existentialist is trying to be social but always ending up isolated and alone, even if you are in a crowd of people, you are still mentally alone. We see this existential behavior in a few different forms. One of which being the fact that his and Marie's relationship is based and ruled solely on physical impulse and physical attraction. When he met Marie, he noticed her physical beauty and reacted to her touch.