So I'm thinking about getting a new graphics card for an HP computer I got last year. However, my computer has an integrated Radeon HD 4200. The card in question that I want to get is the HD 5450, primarily for movies and the occasional game. The 5450's wattage requirement is 300, and my power supply is 300. Would there be any issue?
not unless you're running a Generic PSU with questionable quality.
also note that PSU components age over time and the actual wattage would go down as it gets older.
I also prefer people to have at least 100W more than the PC would ever pull (i.e. at 100% load) that way you aren't running the PSU at max all the time, although companies such as AMD/ATI recommend wattages way over the actual needed wattage as a way to protect consumers from a possible meltdown.
The card doesn't "need" 300 watts, the requirement is your PC will need a 300 watt Power Supply which manufactures will actually overstate somewhat to ensure that even if you have a poorer quality power supply, you should still be okay, so you are fine.