I do not know why Klytemnestra's rage is described as "oily". @Spagirl has a wonderful comment regarding this, upon which I cannot improve:
what I would take from 'oily rage' is that, like oil, her rage will burn at any pressure, but there's a specific point where the pressure is enough to cause a violent explosion.
Klytemnestra is the daughter of Leda and Tyndareus. She and her brother Castor were born at the same time as their half-siblings Helen and Pollux (Polydeuces), who were the children of Leda and Zeus, who had raped Leda by assuming the form of a swan. She is therefore the "daughter of gods and men".
The wound is the death of her daughter Iphigenia. Klytemnestra is wife to Agamemnon, king of Mycenae. Helen in turn weds Agamemnon's brother Menelaus, king of Sparta. When Paris abducts Helen and sets in motion the Trojan War, Agamemnon's fleet is becalmed in Aulis en route to Troy. He is told that he can get fair winds only if he sacrifices something dear to him. He sends word to Klytemnestra that Iphigenia is to wed Achilles. The unsuspecting Klytemnestra sends Iphigenia to Aulis for this supposed wedding, but Agamemnon sacrifices Iphigenia to Artemis. After this, Klytemnestra regards Agamemnon with furious hatred, and takes Ægysthus for a lover. When Agamemnon returns from Troy at the end of the war, Klytemnestra and Ægysthus murder him. She thereby avenges the death of Iphigenia.
Approves is used here in the sense given by the OED as:
To show or prove practically (a thing or person) to be (so and so).
As children of Zeus, Helen and Pollux are immortal. As children of the mortal Tyndareus, however, Castor and Klytemnestra are mere mortals. Had Klytemnestra been immortal, her children would have been immortal as well, and so Agamemnon would not have been able to kill Iphigenia. The death of Iphigenia therefore proves Klytemnestra is mortal. Likewise, her rage and grief at Iphigenia's death shows the strength of her maternal feelings, so this wound "approves Klytemnestra mother and mortal".
"The tide of ill" is nicely put, as tide means both flood and occasion. Agamemnon's murder of Iphigenia led in turn to Klytemnestra's murder of Agamemnon; that in turn led to Orestes's murder of Klytemnestra, and his subsequent torment at the hands of the Furies/Erinyes/Eumenides. All these ills were consequent upon the wound inflicted on Klytemnestra. Note that a Fury is also said to have brought Ægysthus to Klytemnestra's side.
But the tide of ill has its origins well before Agamemnon's filicide. It goes back a couple of generations, to Agamemnon's grandfather Tantalus. Tantalus killed and cooked his son Pelops, then served the meat to the gods to see whether they were truly omniscient. The gods cursed Tantalus to eternet torment and restored Pelops to life, but Pelops himself went on to be cursed by his servant Myrtilus. Myrtilus helped Pelops win a chariot-race for the hand of Hippodamia. Not wishing to share the credit and/or Hippodamia, Pelops murdered Myrtilus. The dying servant cursed Pelops and all his descendants.
Among Pelops's sons were Atreus and Thyestes. The brothers fought over the kingship of Mycenae, becoming bitter enemies. To aggravate matters, Thyestes seduced Atreus's wife. Atreus thereupon killed Thyestes's sons and fed them to the unwitting Thyestes. Determined to have another son who could kill Atreus, Thyestes raped his own daughter, Pelopia. She in turn abandoned the child born of this incestuous rape. The child ended up in the care of Atreus, who raised him as his own. This child was Ægysthus. Atreus's biological sons were Agamemnon and Menelaus, who were thus Ægysthus's foster-brothers.
Atreus sent Ægysthus to kill his mortal enemy Thyestes, but Thyestes recognized him and revealed to him his true identity. Thyestes persuaded Ægysthus to kill Atreus instead, so that they could become the rulers of Mycenae. Ægysthus did so. Thyestes and Ægysthus did rule Mycenae, but eventually Agamemnon drove Ægysthus from power. Agamemnon's murder of Iphigenia and his own death at the hands of Klytemnestra and Ægysthus is the central episode of an entire sordid saga of power struggles, adultery, regicide, and filicide, a saga usually referred to as "the curse on the house of Atreus". This larger tide of ills of which Klytemnestra's story is only a part is set in motion by essentially the same wound: Tantalus's killing of Pelops, like Agamemnon's killing of Iphigenia, is a father's murder of his child as an offering to the gods.
Ægysthus too is swept up in this tide. Poor Ægysthus! In a different telling, he himself could be a tragic figure: born under horrific circumstances, abandoned by his mother, used as a pawn in the enmity between his foster and biological fathers, and seduced by a femme fatale who needs his help to murder her husband. However, no classical source presents him sympathetically, as far as I'm aware. Æeschylus (The Oresteia), Homer (The Odyssey), and Seneca (Elektra) all portray him as weak-willed, simultaneously manipulating Klytemnestra and dominated by her. Pfeiffer has no sympathy for him either, referring to him as "that bearded semblant".
OED has these meanings, among others, for semblant:
a. A person's outward aspect or appearance.
b. esp. as betokening the thoughts, feelings, mood, disposition, etc.
c. The demeanour or ‘countenance’ which a person exhibits towards others.
d. With contextual implication that the appearance is deceitful or misleading.
e. The face, countenance.
Likewise, mawworm is glossed as:
A hypocritical pretender to sanctity.
So in the first place, Ægysthus has a bearded face, quite literally. But he is also untrustworthy. In the Odyssey, Homer writes that Ægysthus slew Agamemnon ὑπο δόλῳ, by deception (III.235). Nestor then explains to Telemachus how Ægysthus seduced Klytemnestra such that in spite of her better judgment (φρεσὶ ... ἀγαθῇσι) she succumbs to his many blandishments (III.266).
As for "fret man's pride", I believe the implication here is that considering how someone as mighty as Agamemnon can be cuckolded and murdered by someone as low as Ægysthus will have a salutary effect on our vanity. When even Agamemnon can be thus humiliated, we should keep ourselves humbled. Alternatively, the phrase could mean that Ægysthus shows us how despicable human beings can be, thereby pricking our vanity.
The poem does not seem particularly adept metrically to me. A couple of the lines are perfectly regular iambic pentameter:
Which gods and men assign the woman here
That bearded semblant, man to outward ken
But the bulk of the lines are so far outside regularity that they're hard to scan. The interest of the poem lies in its theme and structure. Both thematically and structurally, the repetitions of "mother" are very effective: "Mother and mortal", "Mother and spouse", "Mother". By the end of the poem, Klytemnestra sheds her identity as mortal and spouse and becomes just an avenging mother. Her murder of Agamemnon is not just a personal vendetta, however. Klytemnestra is protesting at the narrowness of "the sphere / Which gods and men assign the woman here". Her strike against Agamemnon is also a blow against the tyrants who confine women to such constrained roles: "Woman, thy foot was on thy tyrant then". If the only power Klytemnestra can wield in such a society is "the power to wreak high ruin", then she will seize it.
There is an implicit linkage and contrast here between Klytemnestra and another famously enraged Greek antiheroine, Medea. Trapped and helpless, Medea kills her children when her husband abandons her. Klytemnestra kills her husband when he kills their child. I have no ancient Greek, so I cannot comment on any verbal echoes of Æschylus; but the thematic parallels to Euripedes are clear. Euripedes wrote two plays about Iphigenia: Iphigenia in Aulis and Iphigenia in Tauris, as well as the eponymous Medea. His Medea commits an act of shocking violence, the only way by which she can protest misogynist social strictures that otherwise leave her helpless. Pfeiffer portrays Klytemnestra similarly: it's easy to condemn her, but Pfeiffer makes us sympathize with her rage.