What is the minimum amount I can spend on a credit card monthly and still improve this score. If it is something like buying a pack of gum and paying it off every month that would be ideal.
Any amount is fine, even a pack of gum. Try not to overthink this. Many people advocate spending at least 1% every month yet less than 10% of your limit, but that's pointless except for the month you're actually going to ask a bank to check your credit score (e.g. the month you're buying the house). If you spend $0 some months and $1 or $500 other months, it doesn't really matter. Just use the card sometimes to keep it active and always pay it off in full.
Under this method just how much could I improve my score in a year or 5?
That depends on what factors are currently putting your score in the 630s. Did you have some late student loan payments or otherwise? Are there any defaults? If no, then opening a credit card will likely increase your score quite a bit. If you don't have any negative marks right now, then in 1 year I think it's reasonable that your score could jump by 100 points or more by adding open lines of credit with a perfect payment history.
Do I actually need a credit card to continue improving my score - could I do it with rent or utility payments?
Probably, yes. Of course you can see some improvements without a credit card, but the kind of significant increases you're striving for will most likely require some type of open lines of credit, since a big factor in the FICO score is based on that.
Lastly, I'd like to comment on this statement you made:
I don't want to be in debt again except for a home. I don't care about cash back, miles, or whatever else they sell. I approach spending with if I can't pay for it in cash I can't afford it mindset.
That's a great mindset to have, but don't confuse using a credit card with having debt. They are not one in the same. For many people, having a credit card causes that person to end up in debt, because they don't have a handle on what they can afford, or they lack self control (or both). Based on the fact that you have already built up an emergency fund, and made a comment about wishing to save $300K in cash to purchase a $600K home, I don't think you are the type of person that would be tempted to overspend on a credit card. So with that in mind, I recommend trying to find a good credit card with no annual fee and high cash back rewards. Then you might as well start swiping that instead of your debit card (or using cash) just to get the 1-2% free money. Make sure to set up your CC to auto pay in full every month (or at least auto pay the minimum) so you never have to worry about a late payment fee or getting a negative mark on your credit report. If I'm wrong and you ever find yourself overspending one month just because you're carrying a credit card, then pay it off in full, stop carrying it in your wallet, and revert to just buying that pack of gum every 6 months.