Hey, you have everything you need in your hands. If your server ist running forge you have already come a long way. You said you use putty to access your server. To transfer files you need FTP access with Filezilla or a similar program. This lets you transfer files from your local computer to your server. If you are on windows you can also use WinSCP (https://winscp.net/eng/index.php). I use this to transfer the files to my mineos server.
Be aware, that there are client-only mods that dont run on your server. (like Optifine, DynamicSourrundings, etc)
The way to go is to stop your server and use WinSCP to upload your mod files to the servers mod folder. ( /var/games/minecraft/servers/yourservername/mods.
Then fire up your server and see, if it starts. Check the logs and crash-reports folder for problems. I find it hard to use the mineos server log in the management page, bacause it will stop after a few messages and a modded server will generate a lot of entrys …
If it comes alive you may want to adjust the config files, because the server config files override the clients config.
Hope you get it running, it started like you and have my server running for 2 years now.