My script stops working after I die.?

First of all, you might wanna just do the following:

local player = game:GetService("Players").LocalPlayer
local character = player.Character

It's better to use Player.Character rather than that.

And you might also wanna use UserInputService instead. Make sure that the script is cloned to the player once you die. E.g. place it in the StarterGui

Corrected script:

-- services
local inputService = game:GetService('UserInputService')
local players = game:GetService("Players")

-- https://developer.roblox.com/api-reference/class/UserInputService

-- variables
local player = players.LocalPlayer

local speed = Instance.new('BodyTrust')
wait(1) -- debug, letting the character load

speed.Parent = Player.Character()

-- functions

userInput.InputBegan:Connect(function(input, gameProcessedEvent)
	if(gameProcessedEvent) then return end -- if they're typing in a chat or GUI, do nothing.

	if(Input.KeyCode == Enum.KeyCode.YourKey) then

		speed.Force = Vector3.new(0,0, - 4000)
	end
end)

userInput.InputEnded:Connect(function(input, gameProcessedEvent)
	if(gameProcessedEvent) then return end -- if they're typing in a chat or GUI, do nothing.

	if(Input.KeyCode == Enum.KeyCode.YourKey) then

		speed.Force = Vector3.new(0,0, 0)
	end
end)

-- events

Reminder to place the script in a parent which that reloads the script when you die, like I've stated before, the StarterGui