The Root of the Doctrine of the Trinity
Barth says that revelation is God speaking, which can't be distinguished from God's act of speaking, and so can't be distinguished from Godself. God's Word is God; the revelation of God is God. Scripture and proclamation are based on God's Word, and can become God's word, but they aren't in and of themselves revelation. All of this is summed up by the phrase 'God reveals himself as Lord': because God is Lord in relation to people, when people receive revelation, God becomes their Lord.
The revelation found in the Bible is the self unveiling, imparted to humankind, of the God who by nature can't be unveiled to human kind. God can be seen only when he chooses to reveal himself - we can't find him ourselves. The Bible isn't so much about universal truths as it is a record of specific encounters of particular individuals. The revelations recorded in the Bible are unique and contingent. You can see here that Barth's trying to protect God from any idea that he might be easily graspable. He wants God to be totally other, not someone who's around all the time and who we can too easily get our grubby little mitts on. He's reacting against theology which arguably makes God too accessible, too immanent without much of a sense of God's otherness and transcendence, but it seems to me that Barth just ends up back in the ditch on the other side of the path, probably with a sore bum by now.
Theologians like to use images from the created world, history and culture to talk about the trinity (e.g. the shamrock; a spring-stream-lake; the Old Testament, New Testament and Church; teachers, soldiers and producers in a society). Augustine refers to these as 'vestigium trinitatis' - traces of the Trinity. Barth doesn't like this because he worries that this means we can derive the doctrine of the Trinity from the world rather than revelation, neatly ignoring the fact that, firstly, this isn't really how the idea is usually used, and secondly, that he himself compares God as revealer, revelation and revealedness to the subject, predicate, and object of a sentence. D'oh! Anyway, it's important to him that we don't explain God by the world, but explain the world by God. The Trinity isn't imminent in created things, and created things can't reflect the trinity, but the trinity can reflect itself in created things (a subtle difference, I grant you). To say that creation illustrates revelation is to suggest that revelation is insufficient.
The Triunity of God
The God who reveals himself in Scripture is one in three distinctive modes of being: Father, Son and Holy Spirit. The three persons aren't three Gods. The Father, Son and Holy Spirit means that God is one God in threefold repetition. God's 'personality' belongs to his one unique essence. There aren't three divine 'I's but three times the one divine I. Barth argues that without the doctrine of the Trinity, the concept of revelation ends up sticking a third thing between humankind and God to mediate between them, which isn't ok. The idea of number in God is a metaphor. Barth thinks that talking about 'modes' or 'ways of being' in God is better than talking about 'persons.' He has a point, as we tend to think of persons as implying completely separate physical and psychological entities, which wasn't quite what the original formulators of the doctrine of the Trinity meant. Tritheism is bad. But so, on the other hand is modalism, the heresy which says that God is one but acts in three different ways or modes, ignoring the necessary threeness. It might just be partly the problems of translating from the German, but I can't help wondering if Barth is, once again, in the ditch (Having fun down there, Karl?).