Since the cable isn't run yet, I would definitely buy a new spool of Cat 6, before you start. Even if Cat5 cuts it, it's no good for the future.
As for whether it can connect 1Gbps, I've had mixed success. I have one 6ft cable labelled as "Cat5", no e. House is wired with Cat5e. Some computers/NICs I have used that cable from the wall to the computer, somehow connect at only 100Mbit--somehow it "knows" it's Cat5? Some other NICs have connected at 1Gbps successfully with the same cable but I think to be on the safe side go with at least Cat5e. I've sinced moved that cable for use at work where the router here in my office is only 100Mbit and it doesn't matter. So it's hit or miss whether you'll get a 1Gbps connection. The longer the cable gets it's probably going to be less likely to connect at 1Gbps as well. It's not worth the waste of time installing it all only to find out it doesn't work well (or at all) at 1Gbps.
Again, since you're buying/installing new stuff I would go with Cat6 for sure.