Bettas are warmer water fish than goldies--they will not mix happily. The betta may also have a tough time competing for food, since they are slower swimmers than goldies.
Goldifsh really are not appropriate fish for a 5 gallon tank. They will outgrow it, and since they are large waste producers, you will have problems keeping the water clean. I'd upgrade them to a tank that provides a minimum of 3 gallons per inch of adult fish (don't go based on their juvenile size! Some goldfish can reach 12-18 inches, easily), and then put a heater and the betta in the 5.
Just another comment--you should not add more fish to a tank that is already having problems. Not only will the additional waste load likely increase your problems, but you then expose the new fish to disease...And likely expose your existing fish to new problems as well. Not to sound harsh, but you need to slow down and do some research.
Yup What she said.
The most difficult part of this hobby (for me) is being patient. There are very few of us who haven't gone through the I want more fish in my tank process, and unfortunately a lot of us have killed more fish than we want to admit. Tank size is the biggest limiting factor, and the different needs of different fish i.e. Goldies and beattas also limits what we can and can't do successfully. I would love to keep neons in my Oscar tank, but some things just aren't doable. Good luck, be carefull. eventually you'll get very frustrated if you don't take your time. Many people give up on the hobby after bad experiences, I'd hate to see this.
How many fish do you have in that little 5 gallon tank? That sounds more like what I would put in a 55.... and if it's plecos instead of CAE's or SAE's, I'd be thinking of 300 or so.
I'm not trying to sound harsh or anything, but that's how I think. Did you realize that your goldies will in all likelihood (and proper environment) grow up to be the SIZE of your 5 gallon tank, not just to a size that fits? (Goldfish can be over a foot long at full size and can live as long as a century depending on the breed... although to give credit, most top out at 8-9" and live only 15-20 years....)
I agree with Puma that you should trade your goldies and algae eaters in on a heater and a betta. Your tank will cause you far less hassle for it, and you won't have fish dying all the time.