For a detailed analysis, see St. Thomas Aquinas’, Summa Theologiae, First Part, questions 3-20 or so. He pretty much derives these divine perfections from his five ways of proving God’s existence, which trace God’s action as cause of the universe. He presents a philosophical basis for the doctrine which just is what is based on reason (as opposed to revelation). Walter Farrell’s Companion to the Summa offers a pretty good summary, and Peter Kreeft has a more contemporary Summa of the Summa. I’ll try to briefly summerize the jist of a few:
Omnipotent: God is the first necessary cause of everything that is, that is real, so his power extends to anything that is or could be real. So any thing that can cause anything that can be.
Omniscient: God is the first cause of the ordering of creation, which follows an intelligent design. Therefore, he is himself intelligent. And if he is intelligent, he primarily knows himself, and in knowing himself, he knows what he creates. So he knows everything there is or could be.
Unchanging: God as the only completely necessary being cannot be other than he is since the ability to be other would be a possibilty or potency, and so not necessary, but contingent. Being unchanging, he is outside of time (since time is a function of things changing), so he is eternal.
Perfect: God is the cause of all perfections in the world, so he must have the perfections he causes. He is goodness, truth, beauty. Also, being the first necessary cause, he cannot lack any perfection, because then he would be potential (lacking and able to attain it), and so not completely necessary.
Love: Love is to will good for another, and since God in creating a good creation, loves that creation into being.