Hi,
I am new here.
I am currently a PhD candidate and this is my first round of job applications. One of the jobs I am applying for is actually at the school where I did my undergrad, and the chair of the search committee was my undergraduate advisor. My question is, should I contact this person and let them know I am applying? Should I contact them and say anything else? How should I handle this?
Thanks.

I think contact them and say you are applying. It can't do any harm, unless you didn't have a good relationship with them of course, in which case you better hope they don't remember you!

Contact whoever is listed as the contact inquiry person on the job advertisement. If it's the chair, then contact them. If it's not, I'd be hesitant only in that it might come across as you not being able to follow position description instructions if they've listed a specific person. One thing though is to see whether the listed person is an HR person, or someone else involved with the project. Ideally, they should list someone to tell you specifics about the position itself (i.e. a member of the department) and an HR person about how to apply.

Don't tell them you are applying, say that you are thinking about applying and would like to know more about the role. Have a couple of preliminary questions prepared about the role, i.e. what research background are they looking for, etc. Saying "Hey I'm applying" could come across as you requesting favour...whereas saying "Hey, I'm thinking about applying" comes across more as a general inquiry.

I don't know whether this will have any impact on your application but it could. For example, in May I was sent a job that a number of people thought I'd be interested in. I emailed the inquiry person asking if they'd be open to a brief phone call about the position. They were, so I called them and we had a brief chat. I asked about the project the job was focused on, the kind of research background they were looking for, etc. They asked me a couple of questions about my background.

After that chat, I applied for the position (and got a former mentor to review my application prior to submission, I would recommend doing this!). I landed an interview, and was offered the job two business days after that, which I start in two weeks :)

Thank you, that is extremely helpful!

I looked at the application and there is no mention of where to send questions, unfortunately. The address to send applications is the generic search address where all of the job applications in that department go. Any thoughts?

I could a) email the person I have communicated in the past at that email address with, when I applied for a summer visiting position there, or
b) email my undergrad supervisor, who had originally forwarded the job to the email list which included my master's supervisor, who forwarded the position to me, or
c) just let it go and apply