Make character face mouse.hit.lookVector at all times?

Well, the mouse's lookVector is constantly changing, so just setting the Orientation of the player's HumanoidRootPart will not keep them in sync. Instead, you can try this

First, to check how much the player's mouse will affect movement (higher numbers will make you spin too much), you set _G.MouseSensitivity to 0 (changing this value will make the camera move with the mouse)

Next, we have a few functions to make the mouse move with the camera when not set to 0

Then you will need to check the hit.p of the mouse for the player to move with it (as seen in the Mouse.Move Event)

This LocalScript will move the player to match the mouse, along with some functions to change the camera

_G.MouseSensitivity = 0
_G.MouseSensitivityOffset = 0

local Player = game.Players.LocalPlayer
local Camera = workspace.CurrentCamera
local Mouse = Player:GetMouse()


function RotateCamera(x, y)
    Camera.CoordinateFrame = CFrame.new(Camera.Focus.p) * (Camera.CoordinateFrame - Camera.CoordinateFrame.p) * CFrame.Angles(x, y, 0) * CFrame.new(0, 0, (Camera.CoordinateFrame.p - Camera.Focus.p).magnitude)
end

function GetAngles(cf)
    local lv = cf.lookVector
    return -math.asin(lv.y), math.atan2(lv.x, -lv.z)
end

local LastCF = Camera.CoordinateFrame

function UpdateSensitivity()
    if _G.MouseSensitivity ~= 1 then -- no need to do this if it's 1
        local x, y = GetAngles(LastCF:toObjectSpace(Camera.CoordinateFrame))
        Camera.CoordinateFrame = LastCF
        RotateCamera(x * -(_G.MouseSensitivity + _G.MouseSensitivityOffset), y * -(_G.MouseSensitivity + _G.MouseSensitivityOffset))
        LastCF = Camera.CoordinateFrame
    end
end

Mouse.Move:connect(function()
    UpdateSensitivity()
    local root = Player.Character.HumanoidRootPart
    Player.Character:SetPrimaryPartCFrame(CFrame.new(root.Position, (Mouse.Hit.lookVector*Vector3.new(1,0,1)) + (root.Position*Vector3.new(0, 1, 0))))
end)

Mouse.Idle:connect(function()
    LastCF = Camera.CoordinateFrame
end)

You can change "lookVector" to "p" if you want the player to rotate with the mouse as well I will also note that the "lookVector" doesnt not function properly if the player begins to move, so using "p" is probably the preferred method.

Also, if you don't need to deal with the camera, then only this section is required

_G.MouseSensitivity = 0
_G.MouseSensitivityOffset = 0

local Player = game.Players.LocalPlayer
local Mouse = Player:GetMouse()


Mouse.Move:connect(function()
    local root = Player.Character.HumanoidRootPart
    Player.Character:SetPrimaryPartCFrame(CFrame.new(root.Position, (Mouse.Hit.lookVector*Vector3.new(1,0,1)) + (root.Position*Vector3.new(0, 1, 0))))
end)