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You'll have to be a bit more specific about the connection with the meaning of “eikon” and “charakter”.

It was probably the Hebrew 1:3 reference with the others as variations on the theme.

Again, Elizabeth's declaration, in the context of her being the wife of a Jewish priest, that her cousin Mary was the mother of her Lord is a straightforward declaration at the very beginning of the gospel of Luke that Jesus was Immanuel, God with us, Yahweh, the God of Israel born into human flesh, tabernacling, so to speak, amongst his people as a human (John 1) as he had once dwelt amongst them as a presence in and over the tabernacle in the wilderness (Leviticus 16), now become the image of the invisible God (Colossians), the exact representation of his being (Hebrews), God taking on a human body (I Timothy) and, as Jesus said to Philip who asked him to show him the Father, "Don’t you know me, Philip, even after I have been among you such a long time? Anyone who has seen me has seen the Father. How can you say, ‘Show us the Father’?. (John14)"

It's taken for granted that God by definition is self-existent. Notice what Jesus says about his own nature in the passage below. Also note, in the context of why the Jewish leaders were trying to kill him--his claimed equality with God--his repeated underscoring of this by his use of equivalencies with respect to the Father. Jesus caps it all with a reference to the resurrection and final judgment which was understood by those who were plotting to kill him as something that was going to be effected by Yahweh and claims that it will be at the sound of his voice that the dead will be raised and upon his authority that the good and evil will be judged. This was also a slam against the Sadducees who claimed there was no resurrection: "Yeah, there is. And I'm the one whose going to do it and I'm going to judge you!"

It's actually a pretty funny scene, because just as he did at the healing of the paralytic, he took what was in their hearts and acknowledged it publicly--"Yeah, you're right, who can forgive sins but God alone" and went on to rub their noses in it again and again and again. He didn't say, "Oh, I'm sorry for having given you the impression that I was claiming equality with God. I am but a demiurge."
16 So, because Jesus was doing these things on the Sabbath, the Jewish leaders began to persecute him. 17 In his defense Jesus said to them, “My Father is always at his work to this very day, and I too am working.” 18 For this reason they tried all the more to kill him; not only was he breaking the Sabbath, but he was even calling God his own Father, making himself equal with God.

19 Jesus gave them this answer: “Very truly I tell you, the Son can do nothing by himself; he can do only what he sees his Father doing, because whatever the Father does the Son also does. 20 For the Father loves the Son and shows him all he does. Yes, and he will show him even greater works than these, so that you will be amazed. 21 For just as the Father raises the dead and gives them life, even so the Son gives life to whom he is pleased to give it. 22 Moreover, the Father judges no one, but has entrusted all judgment to the Son, 23 that all may honor the Son just as they honor the Father. Whoever does not honor the Son does not honor the Father, who sent him.

24 “Very truly I tell you, whoever hears my word and believes him who sent me has eternal life and will not be judged but has crossed over from death to life. 25 Very truly I tell you, a time is coming and has now come when the dead will hear the voice of the Son of God and those who hear will live. 26 For as the Father has life in himself, so he has granted the Son also to have life in himself. 27 And he has given him authority to judge because he is the Son of Man.

28 “Do not be amazed at this, for a time is coming when all who are in their graves will hear his voice 29 and come out—those who have done what is good will rise to live, and those who have done what is evil will rise to be condemned.
John 5