I want to start building relationships with influencers/bloggers in my industry; when I interact by commenting on their blog site, on twitter, facebook, etc. should I be using my personal account or should I be using my website brand account?
This really depends on the industry you are in.
Is it not possible for you to do both?
Building your personal brand and also establishing your website at the same time?
The best example I can think of here is how Gary Vaynerchuk started Wine Library TV, but leveraged his own personality which in turn raised awareness for the Wine Library brand.
When commenting on other people's blogs for example, there is no harm in your leaving a comment with your name as [your name] @ [your brand name].
If your website is less established, I'd sway more to starting out with a lot more of a personal approach. Other bloggers are likely to be much more receptive to seeing person actually engaging with their content, rather than thinking "who the hell is this random business who are obviously trying to leverage MY brand?"