Why does the author of Hebrews picture Esau as an immoral and godless person?

The translation you quoted distorts the meaning of the original language. A better interpretation reads,

Lest there be any fornicator or profane person, such as Esau, who for one meal sold his own birthright.

In this passage, he is only presented as being profane, and not as a fornicator. And in this light Esau was profane; the mores of the time expected him to treasure his status (as heir apparent of his household) much more than a bowl of stew.

The idea of Esau's immorality and godlessness is derived from the biblical account of his lack of appreciation of his birthright and his plan to kill Jacob. However, there are also many stories about his evil ways in ancient rabbinical tradition. Although most of these were not preserved in writing until later, many were probably known when Hebrews was written. Among them are:

• Esau had won the affection of his father by lying words (Targ. Pseudo-Jon. to Gen. xxv. 28).

• He was a ne'er-do-well, who insulted women and committed murder, and whose shameful conduct brought on the death of his grandfather, Abraham (Pesiḳ. R. 12).

• Isaac lost his sight from trying not to see Esau's evil deeds (Pesiḳ. R. 12; Meg. 28a; Gen. R. lxv.)

• Esau spent most of his days visiting the shrines of idols. (Gen. R. lxiii.)

• He engaged in blasphemous speeches (Gen. R. lxiii.; Pes. 22b) and in denials of God and the resurrection.

• He is considered as one of history's so three great atheists (Tan., Toledot, 24; Sanh. 101b).

• The sale of the birthright took place while Jacob was preparing for his father who was in mourning for Abraham.

(Above citations from the Jewish Encyclopedia article on Esau)

In addition to the above, we should mention the Book of Jubilees, in which Rebekah dies of grief soon after warning Jacob of Esau's murderous plans and, later in life, Esau later engages in a military assault against Jacob's clan. Jubilees was well known to early Christians, and is mentioned in church fathers such as Epiphanius, Justin Martyr, Origen, Isidore of Seville and others.

Thus, there were many stories about Esau's immorality and godlessness. Although most of them were not preserved in writing until later, it is almost certain that many of them were part of oral traditions available to the author of Hebrews.

The birthright was the passing along, the actualization, of God's promise to Abraham of offspring through Isaac. The offspring in view are both numberless, as the sand, and singular, as the Christ:

Now to Abraham and his seed were the promises made. He saith not, And to seeds, as of many; but as of one, And to thy seed, which is Christ. - Galatians 3:16

Regardless of later trickery on Jacob's part, and Rebekah's collusion, when Esau came home from hunting, empty-handed and hungry, he valued immediate satiation of his physical need over the fulfillment of God's promise to and through him:

“Look, I am about to die,” Esau said. “What good is the birthright to me?”

It is not that godless Esau was hungry but that he despised his birthright, which was to be the channel of God's promise. Compare this to godly Jesus being hungered and tempted in the wilderness. Here Jesus despised his own physical need in favor of the purposes of God.

For they that are after the flesh do mind the things of the flesh; but they that are after the Spirit the things of the Spirit. For to be carnally minded is death; but to be spiritually minded is life and peace. Because the carnal mind is enmity against God: for it is not subject to the law of God, neither indeed can be. So then they that are in the flesh cannot please God. - Romans 8:5-8