Am I using this RemoteEvent correctly?

Well, partially, but you have some problems in your code. Let's walk through this.

First off, make sure that the client isn't trying to fetch an object not created yet. To solve this, make a RemoteEvent manually and put it in ReplicatedStorage, and for even more safety you can add wait(1) at the top of the local script or do something with WaitForChild

Now onto the big issue I see:

Looking in this code I see that you are using "LocalPlayer" which cannot be used in a server script.

Instead, do it like this: (make sure to use :Connect as :connect is deprecated) Also, it is better to put the even in ReplicatedStorage, that way it doesn't get lost in workspace, and can be accessed by the server and client.

local event = game.ReplicatedStorage.SomeEvent;
event.OnServerEvent:Connect(function(player, data) 
       -- Do something with player.
end);

To fire this event:

local event = game.ReplicatedStorage.SomeEvent;
script.Parent.MouseButton1Down:Connect(function()
    event:FireServer(WhatEverDataYouNeedToSendGoesHere)
end)

And there is no need to send the player with the data, as the server already receives that as it's first parameter.

Now, It seems like your trying to make some sort of chat system. I won't go full blown here and recede it all to work for you, plus I don't even know if that's your entire script but here is an example:

Server:

local chat = game:GetService('Chat');
local event = game.ReplicatedStorage.SomeEvent;
event.OnServerEvent:Connect(function(player, message) 
       player.Character.Head.BillboardGui.MainFrame.PlayerLabel.Text = 	
       chat:FilterStringForBroadcast(message, player);
end);

Client:

local event = game.ReplicatedStorage.SomeEvent;
local someTextInput = script.Parent.Parent.SomeTextInput; -- This is an example, you will need to replace this with whatever you are actually trying to grab input from.

script.Parent.MouseButton1Down:Connect(function()
       event:FireServer(someTextInput.Text);
end);
someTextInput.FocusLost:Connect(function()
       event:FireServer(someTextInput.Text);		
end);

This should work provided you know how to finish connecting it, I assume script.Parent is a button and just as an example script.Parent.Parent.SomeTextInput would be the textbox?

My point is that aside from a few issues with actually using RemoteEvents, the main problem is that you are trying to access things from the client server-side, which doesn't work.

My advice is to read this tutorial on how to create a custom chat: http://wiki.roblox.com/index.php?title=Custom_chat_GUI

And read more on RemoteEvents: http://wiki.roblox.com/index.php?title=Remote_Events_and_Functions

It will also be helpful to read about Filtering Enabled: http://wiki.roblox.com/index.php?title=FilteringEnabled

I know this was kind of lengthy but I hope it helps!