HI Bubelaj,
As you have mentioned that you have upgraded from Adobe Photoshop Lightroom 4 to Adobe Photoshop Lightroom 5 and you have received your serial number through chat. Now, you have to install Adobe Photoshop Lightroom 5 to a new machine, you are unable to locate your serial number even on Adobe support page.
I would like to inform you that you would need to follow the instruction mentioned on this link to be able to locate your serial number: - Find your serial number. As you have upgraded from Adobe Photoshop Lightroom 4 to Adobe Photoshop Lightroom 5, I would recommend you to check out this link: - Error: "This serial number is not for a qualifying product" | Adobe Creative Suite, it talks about activating current version using the serial number of a qualifying serial number.
I have also checked your account details in our record and I was unable to find any product or and case via email registered under this Adobe Id. Therefore, I would request you to make sure you to use correct Adobe It while signing in to Adobe.com, maybe to have the serial number and the case registered under a different email address.
Please feel free to update this discussion in case you need clarification with any of the steps mentioned in the links above.
Thanks.