Should I use my "personal" or "website brand" account to interact and build relationships with influencers/bloggers in my industry?

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?"

You need to build a marketing strategy for your personal brand that aligns to your goals. There's a lot of variables that go into it, but I can teach you how to use some models that you can use over and over again to steer your personal/website brand growth. Let's setup a call.

Without question, you need to brand and market both. It's an ongoing effort that will require effort, patience, and time.

I launched our firm in 2009 and at that time no one in the WordPress community new me or my agency. I started marketing the company first, but realized it was much easier to market myself as the face of the company.

Overtime I realized people respected the company and our work, but they related to me the person. The more I infused myself in our marketing efforts, the faster the company branding grew.

Bloggers and social media users in general will support and become brand ambassadors for the company, but they do so because they like the people behind it.

Personally, I would use your personal account when it comes to commenting on their posts and interacting on social sites. They'll be able to tell that you're associated with a specific brand/company, and acknowledge it. While everyone enjoys having brands interact with them, and comments on how smart and on top of things they are, the reality is that individuals want to connect with other individuals, and will like the brand by association with that individual.