Does God tell us in the bible he is omnipotent and omniscient, or are these attributes which (Greek?) philosophers and theologians merely ascribed to him?
A simple Bible search for 'almighty' would have given you more than enough passages about omnipotence. The first mention, for example, is:
Genesis 17:1-2 (ESV)
1 When Abram was ninety-nine years old the LORD appeared to Abram and said to him, "I am God Almighty; walk before me, and be blameless, 2 that I may make my covenant between me and you, and may multiply you greatly."
For omniscience, similarly many, many passages exist -- the combination of them shows quite strongly that God is omniscient. For example:
1 Samuel 2:3 (ESV)
3 Talk no more so very proudly,
let not arrogance come from your mouth;
for the LORD is a God of knowledge,
and by him actions are weighed.
1 Samuel 16:7 (ESV)
7 But the LORD said to Samuel, "Do not look on his appearance or on the height of his stature, because I have rejected him. For the LORD sees not as man sees: man looks on the outward appearance, but the LORD looks on the heart."
Matthew 6:4 (ESV)
4 so that your giving may be in secret. And your Father who sees in secret will reward you.
Acts 2:23 (ESV)
23 this Jesus, delivered up according to the definite plan and foreknowledge of God, you crucified and killed by the hands of lawless men.
He is definitely almighty. There are many psalms that praise his power, and say that there is no one like God. Revelations states:
And I heard as it were the voice of a great multitude, and as the voice of many waters, and as the voice of mighty thunderings, saying, Alleluia: for the Lord God OMNIPOTENT reigneth.(Emphasis my own)
God, is definitely all-knowing. He knows the hearts and intents of men. He knows what they are going to do. In Matthew, it shows that Jesus is able to discern the thoughts of people:
And Jesus knowing their thoughts said, Wherefore think ye evil in your hearts?
Others can provide a more comprehensive list of Scripture, but time and time again in the Bible, it has been made quite clear that God is supreme and divine, knowing all, having power over all, and is everywhere. He is infinite.
Jesus proclaims to John in the book of Revelation His own might and glory. Also, 2 Timothy 3:16, "All scripture is inspired by God (literally "God Breathed")..." That is why they quote other scripture because it is God using the words He gave to those who wrote the books to declare who He is to us.
10 And he said to me, “Do not seal up the words of the prophecy of this book, for the time is near. 11“Let the one who does wrong, still do wrong; and the one who is filthy, still be filthy; and let the one who is righteous, still practice righteousness; and the one who is holy, still keep himself holy.”
12“Behold, I am coming quickly, and My reward is with Me, to render to every man according to what he has done. 13“I am the Alpha and the Omega, the first and the last, the beginning and the end.”
14 Blessed are those who wash their robes, so that they may have the right to the tree of life, and may enter by the gates into the city. 15Outside are the dogs and the sorcerers and the immoral persons and the murderers and the idolaters, and everyone who loves and practices lying.
16“I, Jesus, have sent My angel to testify to you these things for the churches. I am the root and the descendant of David, the bright morning star.”
In Jeremiah 17:10 He also claims to have the power to know every mind and heart, and not only that but to have the power and authority to judge the ways and deeds of that man. “I, the LORD, search the heart, I test the mind, Even to give to each man according to his ways, According to the results of his deeds."
Also, 2 Timothy 3:16, "All scripture is inspired by God (literally "God Breathed")..." That is why they quote other scripture because it is God using the words He gave to those who wrote the books to declare who He is to us.
The clearest and most straightforward references to my mind are:
John 1:1-4 pointing to creation of all that exists
Genesis 1:1 as above
Hebrews 1:3 upholds all things
Being the creator of the universe and everything that exists, makes him omnipotent. Nothing exists that he did not create, and the creator owns, and is above, his creation.
Ecclesiastes 12:14 every single action of man will be judged. God is aware of all actions by all humanity.