This should work. Add a file /home/username/.config/systemd/user/my_cool_service.service with this content:

[Unit]
Description=My cool service

[Service]
Type=oneshot
ExecStart=/home/username/my_cool_script.sh

[Install]
WantedBy=default.target

Now add the script /home/username/my_cool_script.sh.

#!/bin/bash
echo "Hello from my cool script."

Enable and run the service.

$ chmod +x /home/username/my_cool_script.sh
$ systemctl --user daemon-reload
$ systemctl --user enable my_cool_service.service
# Optional:
$ systemctl --user start my_cool_service.service 
$ journalctl -e --user-unit=my_cool_service # You should see the echoed string from the script.

The service should now run every time the user username logs in.