If I wanted to promote Christianity...?
There is nowhere to begin without the Jewish canon, but to begin with why label it 'Old'. Perhaps you labeled the Jewish scriptures as 'Old', so that you could suggest you have a newer better testament? That is how it seems, but that is not actually what the label is intended to be for. 'Old' is not intended to mean 'Antiquated', however this is what it connotes in modern language, and it is what many people including many preachers will be happy to assume. I think that calling it 'Old testament' is something like sawing of the branch one is sitting upon.
As an aside: Christianity is not meant to be promoted. That is counter productive. Promotion of it evidences a problem, since it should simply draw everyone to itself. It is presumed in the gospels that someday Christianity will do this effortlessly. There is a suitable figure of speech in the gospels that a city set on a hill cannot be hidden, but many people have lost touch and think that preaching means promoting Christianity through announcements, arguments and discussions. This is not the case, and that is not preaching. Promotion is not preaching. If the movement is to be successful it must return to basics, moving beyond these rookie mistakes.
why would I use the Old Testament as a source?
You'd use the Tanach as a source, because it is grounded in philosophy and experience. Without it there is nothing to discuss. It is the connection between humanity and the experience of the Jews. I stop short of saying it is obfuscated. Its not obfuscated so much as not explained to outsiders (or it perhaps cannot be explained to outsiders). Whether it is literal or not it is a slowly developed body of wisdom that is passed from parent to child for many generations before being written down, being nearly lost. Before its writing (presumably by Ezra) it is taught by word of mouth, parent to child for generations. The parents of multiple generations pack into it a lot of wisdom and direction, and the way they teach it is very personal and philosophical. The parent-child relationship is the context for the tanach, and outside of that parental relationship the tanach begins to fall apart in its meaning and relevance.
You and I are outside of that parental situation. That is where the NT writings come in, which partly crack it open for outsiders. When Jesus opposes murder it is because that is an interpretation of the Tanach and its message. It is not like the NT completely connects us to the tanach though. There is a lot that is not explained and which Christians get geeky about sometimes. There is a lot of pleasure in puzzling it out. To ask why you'd use Tanach as a source is really like asking why there are Christians, because without the tanach there can be none. If you erased it from the world you'd erase Christianity, too.
Hence the label 'Old Testament' is ironic, since its still needed in modernity.
On cursory looking it seems to involve a lot of violence from various quarters. Shouldn't I try to ignore that half of the bible and hope no-one notices? Put me straight!
Think of reality as most real in the middle and less real as you go towards the outer layers. That is I think the way to interpret the Tanach and probably how it is intended. The centre of the tanach is the law, and everything else is less important: relatively less real. If you depart from this, then it makes little sense. When reading
Genesis focus on laws, principles, morals and guidelines. If the law says not to murder, then you must not murder end of story even if Cain does. At least that is how parents likely teach it to their children. Obviously some outsiders such as ourselves have grabbed the translated stories and used them to promote violence, but this does not mean Christians can simply get rid of these precious stories. They are our connection to the philosophy of the Jews. Instead Christians must focus on principles of nonviolence and put the actions in these stories under judgment philosophically using the law. Analyze them using the law as if we are being taught by a parent. "Was it Ok for David to kill goliath? Why or why not?" "Was it really Ok for Sampson to kill the Philistines? Why or why not?" "Was it Ok for the LORD to drown the whole world?" In all three cases my answer is 'No'. Its not Ok to drown the world, not Ok to kill Goliath, not Ok to crush the Philistines. Law and morality are the actual context of these stories. This creates an ironic situation and should cause you to question the literal stories. What really happens with Noah? What is the 'Flood'?