The distinction between mortal and venial sins is only very roughly that "venial sins are kind of bad and mortal sins are really bad". More accurately one could say that all sins adversely affect one's relationship to God; but venial sins, because they don't involve a complete turning away from God, "merely" damage the relationship, while mortal sins destroy it and require a new outflow of grace from God—a "reboot" of the relationship.
As far as finding a Biblical basis for this distinction: The first thing to understand is that given an arbitrary Catholic belief, there may or may not be a Biblical basis for it. In this case, however, we can find at least something to go on.
There is one quotation which the Catechism of the Catholic Church cites as underlying the belief in a distinction between mortal and venial sins:
If anyone sees his brother sinning, if the sin is not deadly, he should pray to God and he will give him life. This is only for those whose sin is not deadly. There is such a thing as deadly sin, about which I do not say that you should pray. All wrongdoing is sin, but there is sin that is not deadly.
(1 John 5:16–17; emphasis added)
The Catechism specifically states that "the distinction between mortal and venial sin, already evident in Scripture, [citing the quote above] became part of the tradition of the Church." It goes no further in supplying any scriptural support. The Summa Theologica of Aquinas, however, has an extensive discussion of sin (First Part of the Second Part, Questions 73 and 74—titled "Of The Comparison Of One Sin With Another" and "Of The Subject Of Sin") in which Aquinas concludes (among other things) that there are reasons for believing that certain sins are judged less grievously than others, either because of the kind of sin or because of the circumstances. Among the quotations Aquinas gives are 1 Timothy 1:13:
I was once a blasphemer and a persecutor and an arrogant man, but I have been mercifully treated because I acted out of ignorance in my unbelief.
Aquinas also quotes John 19:11:
Jesus answered [him], "You would have no power over me if it had not been given to you from above. For this reason the one who handed me over to you has the greater sin."
and interprets this (as the Church appears to have always interpreted it) as saying that Judas' sin is greater than Pilate's.
I see that my original answer to the question does not address your specific query about the Biblical basis for the belief that "the knowledge/deliberateness of one's sin impacts its effect on the relationship between the sinner and God." The Catechism addresses this question indirectly, stating that "freedom makes man responsible for his acts to the extent that they are voluntary" and "every act directly willed is imputable to its author". As an instance of the deliberateness of an act playing a part in its evil, the Catechism cites 2 Samuel 12:7–15 (the story of the prophet Nathan's confrontation with David after the arranged killing of Uriah). But as far as knowledge or deliberation making an act worse, the Catechism states that
sin committed through malice, by deliberate choice of evil, is the gravest
but it offers no direct Scriptural support. Nor can I find anything direct in the Summa. But Aquinas, in his Catena Aurea ("The Golden Chain"), an edited collection of commentaries on the Gospels, quotes the commentaries of John Chrysostom and Cyril of Alexandria on Luke 12:47–48:
That servant who knew his master’s will but did not make preparations nor act in accord with his will shall be beaten severely; and the servant who was ignorant of his master’s will but acted in a way deserving of a severe beating shall be beaten only lightly.
(New American Bible, Revised Edition)
Chrysostom's commentary, as cited by Aquinas, is
For all things are not judged alike in all, but greater knowledge is an occasion of greater punishment.
To this Cyril of Alexandria adds
For the man of understanding who has given up his will to baser things will shamelessly implore pardon, because he has committed an inexcusable sin, departing as it were maliciously from the will of God, but the rude or unlearned man will more reasonably ask for pardon of the avenger.
Thus it appears that the doctors of the Church have commonly interpreted knowledge and deliberation as adding fault to the sin.