How can I make this projectile fire in the right direction?

If you want your projectile to fire in the direction the player's torso is facing, which is what it seems like, you were very close. When you had:

bv.Velocity =  root.CFrame.p * 150

The projectile DID spawn, it was just moving so quickly you couldn't catch a glimpse of it, because you used position, which is usually a large value, so your projectile went shooting off at the speed of sound. What you want is lookVector, which is the unit vector of the direction a CFrame is facing.

bv.Velocity =  root.CFrame.lookVector * 150

Enjoy :)

Also, a tip: When using Instance.new(), you can assign a parent instantly, without using another line. Just put a comma, and then what you want to parent the new object to.

 local bv = Instance.new("BodyVelocity", player.Character.Torso)

--Instead of 

local bv = Instance.new("BodyVelocity")
bv.Parent = player.Character.Torso

Here's an idea

1. Make the projectile face to where the mouse is

projectile.CFrame = CFrame.new(projectile.CFrame,(mouse.hit.p.x,mouse.hit.p.y,mouse.hit.p.z))

1. Make the BodyVelocity go to where the projectile is facing

If you want the BodyVelocity's velocity to "face" your mouse.

Projectile.MaxForce = Vector3.new(1e8,1e8,1e8)
Projectile.Velocity = (mouse.hit.p - root.Position).unit * bulletSpeed

To make the projectile face your mouse. (Makes the projectile look nicer)

Projectile.CFrame = CFrame.new(Projectile.Position, mouse.hit.p)

You can get the mouse using the player got from the RemoteEvent.

Here's my gun's script. Don't blame me for untidy lines, I just wanted to make it quick.

--Defines
local tool = script.Parent
local fire = tool.Fire
local remote = tool:WaitForChild("RemoteEvent")

--Settings
local damage = 40 -- Headshot damage is * 2.5
local bulletSpeed = 500 -- Don't go for too much please

--Startup
if tool.Parent:IsA("Model") then
    if game.Players:GetPlayerFromCharacter(tool.Parent) then
        for i,v in pairs(tool:GetChildren()) do
            if v:IsA("Part") or v:IsA("BasePart") then
                v.CanCollide = false
            end
        end
    else
        for i,v in pairs(tool:GetChildren()) do
            if v:IsA("Part") or v:IsA("BasePart") then
                v.CanCollide = true
            end
        end
    end
else
    for i,v in pairs(tool:GetChildren()) do
        if v:IsA("Part") or v:IsA("BasePart") then
            v.CanCollide = true
        end
    end
end
--CanCollide stuff
tool.AncestryChanged:Connect(function()
    if tool.Parent:IsA("Model") then
        if game.Players:GetPlayerFromCharacter(tool.Parent) then
            for i,v in pairs(tool:GetChildren()) do
                if v:IsA("Part") or v:IsA("BasePart") then
                    v.CanCollide = false
                end
            end
        else
            for i,v in pairs(tool:GetChildren()) do
                if v:IsA("Part") or v:IsA("BasePart") then
                    v.CanCollide = true
                end
            end
        end
    else
        for i,v in pairs(tool:GetChildren()) do
            if v:IsA("Part") or v:IsA("BasePart") then
                v.CanCollide = true
            end
        end
    end
end)

--Functions
local function getChildrenWithClassProvided(index, class)
    for i,v in pairs(index:GetChildren()) do
        if v:IsA(class) then
            return v
        end
    end
end

local function onRemoteFired(plr, cmd, value1, value2)
    if cmd == "fire" then
        -- value1 is end point, value2 is the spread
        -- Yay realistic bullet xd
        local root = plr.Character.HumanoidRootPart
        local startPos = fire.Position
        local bullet = Instance.new("Part")
        bullet.Parent = game.Workspace
        --Properties
        bullet.Anchored = true
        bullet.CanCollide = false
        bullet.BrickColor = BrickColor.new("New Yeller")
        bullet.Material = Enum.Material.Neon
        bullet.Size = Vector3.new(0.25, 0.25, 2.5)
        bullet.CFrame = CFrame.new(startPos, value1)
        bullet.Transparency = 1
        bullet.Name = "Bullet"
        --Realistic effects
        fire.smoke.Enabled = true
        fire.flash.Enabled = true
        fire.light.Enabled = true
        wait(0.1)
        fire.smoke.Enabled = false
        fire.flash.Enabled = false
        fire.light.Enabled = false
        bullet.Transparency = 0
        bullet.Anchored = false
        local bulletLight = Instance.new("PointLight")
        bulletLight.Brightness = 10
        bulletLight.Range = 6
        bulletLight.Color = Color3.new(255, 255, 127)
        bulletLight.Enabled = true
        bulletLight.Shadows = true
        bulletLight.Parent = bullet
        bulletLight.Name = "Light"
        --Movement
        --A lot of physic and math stuff that will get your brain explode
        local mouse = plr:GetMouse()
        local direction = (value1 - fire.Position).unit
        local spawnPos = fire.Position + direction * 2
        bullet.Velocity = direction * 100 -- yay more math
        bullet.CFrame = CFrame.new(bullet.Position, mouse.hit.p)
        local bodyForce = Instance.new("BodyVelocity")
        bodyForce.Parent = bullet
        bodyForce.MaxForce = Vector3.new(1e8,1e8,1e8)
        bodyForce.Velocity = (mouse.hit.p - root.Position).unit * bulletSpeed
        bullet.Touched:Connect(function(part)
            if part.Parent:IsA("Model") then
                local enemyhum = getChildrenWithClassProvided(part.Parent, "Humanoid")
                if enemyhum then
                    if enemyhum ~= tool.Parent.Humanoid then
                        local dmgpt = damage
                        if part.Name == "Head" or part.Parent:IsA("Accessory") then -- That hurt .-.
                            dmgpt = dmgpt * 2.5 -- Instant kill
                        end
                        enemyhum:TakeDamage(dmgpt)
                        bullet:Destroy()
                    end
                end
            end
        end)
       game:GetService("Debris"):AddItem(bullet, 2.5) -- Remove bullet if it's here too long
    end
end

remote.OnServerEvent:Connect(onRemoteFired)

It's unfinished so don't try.