In Matt. 5:28 the Greek word used is epithymeo which comes from the root thymos. Here is a study of both the combined word and the root from the Theological Dictionary of the New Testament. I hope you find it helpful.
thymós (→ orgḗ. thýō denotes violent movement (of air, water, the ground, or living creatures). From the sense to boil up comes to smoke and then to sacrifice. thymós means what is moved or moves, i.e., vital force, and it may then denote such varied things as desire, impulse, spirit, anger, sensibility, disposition, and thought. In the NT thymós occurs five times in Paul, once in Hebrews, twice in Luke, and ten times in Revelation (five with toú theoú). The meaning is always wrath, human in Paul, Hebrews, and Luke but not in Rom. 2:8, divine in Revelation except in 12:12 (the dragons wrath). Symbols are wine and cup (from the OT), as well as vials (cf. Rev. 14:10, 19; 15:7). In Rev. 14:8; 18:3 Babylon has brought the nations into ungodliness, so that they have fallen into sin and under Gods wrath (cf. Jer. 25:15ff.). thymós is the same as wrath, and the phrase the thymós of wrath (orgē
occurs in Rev. 16:19; 19:15.
epithymía, epithyméō (→ hēdonḗ ). From where the English word hedonism is derived.
1. This group denotes desire, especially for food or sex. This desire is morally neutral at first, but philosophy, holding aloof from the sensory world, regards it as reprehensible, and in Stoicism epithymía is one of the four chief passions. Epicurus distinguishes between natural and illicit desires, subdividing the former into the purely natural and those that are necessary to happiness.
2. The OT condemns the evil will as well as the evil act, e.g., coveting. Sexual self-discipline is demanded (cf. 2 Sam. 11:2). In Judaism fasting, regulation of meats, and strict keeping of the sabbath are important. Desire is sometimes viewed as the main sin, for epithymía conflicts with supreme devotion to God.
3. Stoicism and Judaism combine in the Hellenists. The LXX has epithymía for base desire in, e.g., Num. 11:4 (though epithymeín can denote pious striving in Is 58:2). Philo summons us to conflict with epithymía.4 Maccabees demands the rule of reason over impulse and numbers epithymía (arising out of sensuality) as one of the chief impulses alongside hēonḗ, phóbos, and lýpē.
4. Rabbinic equivalents are to much the same effect, e.g., in such phrases as doing ones desire.
B. The Usage in the NT. The group is more common in the epistles than the Gospels. It may denote hunger (Lk. 15:16), longing (Lk. 22:15), or a desire for the divine mysteries (Mt. 13:17) or for anything good (Phil. 1:23; 1 Tim. 3:1). But it usually denotes evil desire as indicated by the object (a woman in Mt. 5:28, other things in Mk. 4:19), by the orientation (cf. Gal. 5:17), by the instrument (the heart in Rom. 1:24, the body in Rom. 6:12, the flesh in Eph. 2:3, the eyes in 1 Jn. 2:16), or by the manner (carnal in 1 Pet. 2:11, worldly in Tit. 2:12, defiling in 2 Pet. 2:10, etc.). A Jewish model for Pauls use of the term for the tenth commandment is found in Rom. 7:7. Hence one need not postulate Stoic influence except perhaps in 1 Th. 4:5. In any case, disobedience, not irrationality, is the evil in epithymía. Desire is a manifestation of sin. Under the prohibition of the law, it discloses our carnality, our apostasy from God, and our subjection to wrath (Gal. 5:16; Rom. 1:18ff.). In James it is the root of individual sins (1:1415), while in John it arises out of the world, constitutes its nature, and perishes with it (1 Jn. 2:15ff.). NT statements about epithymía belong to the message of repentance and self-denial. epithymía is impulse, lust, or anxious self-seeking. It shows us what we really are. Even after reception of the Spirit, it remains a danger.
Kittel, G., Friedrich, G., & Bromiley, G. W. (1995). Theological Dictionary of the New Testament (339340). Grand Rapids, MI: W.B. Eerdmans.