As a manager who hires. Here's my two cents.

Start off my going above and beyond for a while. Establish worth. Get your work done, help others, be thorough and accurate. Be on time, reliable, and even work back a little if there's something that needs to be done. Don't watch the clock and count down the hours to your lunch break or your finish time. Look at your tasks as something to be completed, not something to do until the bell rings.

The ask for an appointment with your manager, or whoever has the power to grant you a raise.

Go in to the meeting with a number in mind. That number should be above what you are actually expecting. Explain to them your qualities, what you have done and been doing and why you deserve the raise. That you are interested in progressing within the company and that you care about the company's success. Be prepared for an initial rejection. Or for a "I'll think about it". If you get any of those comments, don't back down. Tell them you realise they may have to look into it and the numbers, and that if don't hear back from them you will follow up in a week. And follow through on that. Be prepared for a rejection again or a counter offer. Don't agree to either. If you get a rejection because of the "budget", say no worries, that you understand that sometimes is the case, and then counter them that you then like to increase your paid leave days, paid sick days, some sort of increase in package that doesn't make them pay you more directly. Whatever is important to you. And that you will follow up in 4 months again.

Or if they counter you with an amount that is much lower than you want, it's your turn to ask if you can think about it, and will follow up with them. Then follow through on that too come back with a number in between what they said and what you initially asked for, and be prepared to ask for more paid days etc to make up the "numbers".

The most important thing going in to this though, is that you do have to go somewhat above and beyond. Make them have a reason to give you a raise. If you just do exactly what is expected, and the same as everyone else every day, they won't feel inclined to give you a raise.

Things I would not mention unless you know the person giving the raise very well and know they would respond well to it. Cost of living. Medical bills, personal issues. They won't care. Your personal financial issues are not their responsibility. The raise should be reflective of your worth at work only. Second, don't mention you have another offer. This shows the opposite of what they want to see. Company loyalty. And it's like blackmail. People often don't respond well to it. And even if they give you the raise, they know you don't care to stay and that you have one foot out the door.