|Is the issue with my computer?|
Well, consider what you said just before that, "the only way to get it back is to unplug the modem and plug it back in. I still get an IP from my router but the router loses its IP."
Does that sound like your PC or the router? To be that says the router is the issue.
Considering you've already recycled power on the router and it's still doing it it seems very likely to me your router is dieing and needs to be replaced.
Before you do replace it, here are two more things to try first:
1) Update the firmware on the router to the most recent
2) Reset router to factory defaults (ie: with unit on, hold reset button in for 10 seconds and release it) and reconfigure it.
If neither one fixes it, it's time to replace your router.