I think a raised bed, if you can irrigate it, should be safe right up to the driveway. It's a light colored drive, which will warm up, but not near to the extent that blacktop will, and it has the benefit of reflecting light onto the undersides of leaves. Leaves have more concentrated chlorophyll above than below, but the undersides are still capable of absorbing quite a bit of additional light, greatly benefitting the plant...
... Back on topic :P, I would leave some space between your raised bed and the neighbors fence, perhaps 18" or so, to walk in. That is nice for the neighbors in case they decide to alter the fence somehow, they won't be worried about walking in your raised bed or damaging your blackberry bushes. Also, I'm assuming you'd plant the blackberries in the back, leaving space behind them is good, as they can spread by suckers, and may appear in the neighbors' yard unless a barrier of some sort is installed.
If that fence is from treated lumber (probably not, by the look of is, but better safe than sorry), or even a previous fence, the surrounding soil may be contaminated with arsenic, which is absorbed by plants and can pose a health hazard. Other than a root barrier, you could get a sample tested for arsenic, and go from there.