What version of Mint are you using?
With Mint 21 (which is based on Ubuntu 22.04) I could use one of the pre-built wheels in Index of /wxPython4/extras/linux/gtk3/ubuntu-22.04 just by running the following commands:
URL=https://extras.wxpython.org/wxPython4/extras/linux/gtk3/ubuntu-22.04
/usr/bin/python3 -m pip install --user -f $URL wxPython
I have just installed Mint 22 on one of my PCs. Mint 22 is based on Ubuntu 24.04 for which there are currently no pre-built linux wheels, or for Python 12 which is standard on Mint 22.
I ran the following command on the PC:
pip install --user --upgrade wxPython
It took some time, but it did complete without any errors being displayed.
I was then able to successfully run some simple wxPython apps, but each time I closed the app there a was a ‘Segmentation fault (core dumped)’ message in the terminal.
This is a known problem with Python 12 that also affects macOS and Windows (see WxPython segmentation fault on Python 3.12 with 4.2.1 on macOS · Issue #2455 · wxWidgets/Phoenix · GitHub). The problem has been fixed in the wxPython GitHub repo, but it needs @Robin to make a new official release (see Release 4.2.2 · Issue #2553 · wxWidgets/Phoenix · GitHub).
As a workaround, it may be possible to use the wxPythonMeticy build provided by @jmoraleda