So I'm in gold 5, and I don't have a gold border like the other people I know have. I don't know if I should go to a higher rank gold or what How do I get the gold border?
You will get your border at the official end of season 4. This will happen on November 11, 2014 as Damek has already pointed out in his answer. Like every season, you will only get the border of your Ranking in a specific Queue. Any change of your ranking that happens after this date will be ignored. This means if you are Gold 1 and in the promotion to platinum you won't get the Platinum rewards if you reach Plat on November 12.
This basically means: If you are gold in Twisted Treeline ranked teams, your border will not show up in Summoners rift (Same goes the other way around). However your highest rated border will be shown on the Crystal Scar and on The Howling Abyss.
Please note that you will not recieve your Rewards if you are chat-restricted or otherwise banned/often reported.
Also your ranking in previous Season has absolutely no effect on the rewards of Season 4. For example (as Philipp pointed out in his comment) If you were Platinum in season 3 and Gold in season 4, your platinum Border will be replaced with a Gold border. Of course you still have the Permanent rewards like the Victorious Skin and the Summoners Icon.
The border you have on the loading screans shows the division you were in in the last season, not the current season.
Last season I Finished the season in Silver V so right now I have a silver border even though I am in gold V now. When the season ends on 11 November I will get the golden border.
The 2014 season ends on November 11, 2014 at 12:01 AM PST
Read this page for more information.
Edit:(thanks to Jeykey)
The border displayed is from the divison you were in last season. If you got to Diamond in S3 you've had an Diamond border in Season 4. If you only managed to get to Platinum in S4 you will get a Platinum border next season.