I established a fixed price with a client for a website design project. I got underpaid because I couldn't show any experience in the area of the project since I didn't had permission from my other clients to show my jobs for them on my portfolio. I accepted this just because the client told me there will be more work up ahead. Now, we're near the end (the site is up and ready) and I've already been paid, and still the client wants to add more features to the home page of the site. How can I tell my client that we are already out of scope for the agreed payment amount and that implementing this new things means that I'll have to charge him for it?
You stated it...
[Client], we are already out of scope for the agreed payment amount. Implementing [these] new things means that I'll have to charge [you] for [them].
There's no "super-secret" language to use. No way to explain a price increase that a client won't see as a price increase. Be forthright, honest, and upfront about it. If you are running a freelance business then run a business. You do not have to do anything a client asks you to unless you are contractually obligated. That includes doing more work than a contract states. If you have reached a contracts completion and you have been paid. Then a new contract may be warranted, especially if the prior contract had no terms regarding updates and maintenance of a web site.
client told me there will be more work up ahead
This is very often a tactic used by bad clients to get you to lower pricing. It's rarely true. And if the client happens to have more work, they will want a lower price for that as well.
NEVER lower your pricing simply because a client wants you to. Not ever.
You can show the contact or the project agreement that the client has shared. If the agreement clearly mention the scope of the project and it has completed then the client has to pay extra for the work which is outside the scope.