Life advice please

There are just some things school and life don’t teach you how to approach, such as asking for a raise.

Some facts:

I’m going to be celebrating my one year at my current company soon, and in the past year I’ve taken on much more responsibilities that are beyond what I was originally hired for. Also in the past year, I saw a $2.50 “raise,” which while nice, isn’t really all that considering that I was already (and still am) underpaid by about $10k according to the going market rate and that I went from a “temp” to regular employee. I also missed out on the company “market rate adjustment” because of the aforementioned “raise.” Additionally, since I assist with compensation and benefits, and have access to all comp data and process 90% of HRIS input, I’ve seen what many other’s have made in other positions outside of my own. Additionally, one of my coworkers left recently, and I absorbed much of what they were resonsible for in addition to my own. And lastly, while I’ve complained about the pains of working for the company, I’ve grown to... tolerate them, make a name for myself, develop strong bonds with the employees we service and many of the directors, provide a level of customer service that I’ve been told exceeds those that had held the position before me, and contribute to driving efficiency and cost savings, and grown from a recent college student entering the workforce full time to a growing seasoned professional.

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Now, with that said, I’d like to make more. I’m currently interviewing for a company and everything seems to sway towards me receiving an offer, but I don’t want to leave my current org just yet... so, I’d like to ask for a raise that equates from 8 to 10k more - which woukd bring me to just about the going market rate. How do I go about doing so? Your advice is welcomed. I work in HR, and I don’t even know how to approach it.

Side note: it’s a stingy company when it comes to pay, but I’ve seen other departments give out increases of about the same, if not greater amounts.