@Rubbello44 You don't looking a way "how to update GT5", but "how to update games"... Download patches from Sony servers, then install all packages by "Install Packages" option in XMB. In order from smaller version to highest You want (some games have only one all-in-one patch, some have many, depend how many is written in seeking XML by updater).
Some games have their own patching system, and packages from Sony are only installers. Such game is i.e GT6. I don't remember if GT5 also have such cancer system, but if yes, then You must install all and at least once launch the game and wait until she install patches.
as mentioned, you have to install every single one of the update packages in release order
On GT5 I made a incremental package from 2.01 to 2.11 so if you have the american XL version you only install this package and the 2.11 on the topic and you can install the mods in full glory: https://gbatemp.net/threads/gran-turismo-5-xl-updates.586262/
You dont need to get past 2.11 to install the mod. The reason is that after 2.11, GT5 didnt had any content updates, only security ones which makes modding (especially on savefiles) more limited.
(I tried doing the same on GT6 which is the absolute worst when updating and unfortunately, it always crashes the game. What I could do however is to make a PKG based on the game data with the updates installed instead of a cumulative update package, but it is very difficult to share a 16GB pkg for free online and I have only the Ayrton Senna version which the SKU code is different)