From this we can deduce that the existence we know of is in some way related to, or an expression of, the nature of this being or cause. How could it not be? as like with an artist the cause would leave a signature or stylistic technique in its creation. Not just an objective signature, but a subjective signature.
The problem comes in with the "signature or stylistic technique" part. I don't think we can distinguish between the necessary and the stylistic. So, for example, if Da Vinci is working in paint, he's bound by the physical facts of paint. How are we to distinguish between the material of creation and the stylistic choices made by God?
The reason I think this is largely impossible is because we don't yet have another creation to compare ours to. I can tell which artist created something only if I have some experience in comparing one painting to other, different paintings. I don't know ahead of time what colors limit the palette and what might have been a result of God's preferences.
If we dodge the limits of the material (the palette) and take our God to be all-powerful (and the other attributes) we are still stuck. It does little good to view a work from an artist who is all artists -- any style is only a subset of the possible styles and no information is obtained by viewing a single work.
I can't even get past the assumption that human beings matter to God. Judging by what we already know, we are just as likely to be a short step on a very long evolutionary staircase. Or, worse for us, we might be on the extreme fringes of His real work -- something happening a gazillion light years away.