How do I use CFrame to rotate a part with the player?

There were a few things I noticed. For each thing I added or changed, I included a comment (*new*). Anything from your original code that I removed, I left but commented out.

local RunService = game:GetService('RunService')
local Player = game.Players.LocalPlayer
local PlayerName = Player.Name
local Character = Player.Character -- *new*
repeat -- *new*
    Character = Player.Character -- *new*
    wait -- *new*
until Character ~= nil -- *new*

local Part = Instance.new("Part")
Part.Parent = game.Workspace
Part.CanCollide = false
Part.Anchored = false -- *new*

local BodyPosition = Instance.new("BodyPosition")
BodyPosition.MaxForce = Vector3.new(math.huge, math.huge, math.huge)
BodyPosition.P = 1000
BodyPosition.D = 100
BodyPosition.Parent = Part

local BodyOrientation = Instance.new('BodyGyro')
BodyOrientation.MaxTorque = Vector3.new(math.huge, math.huge, math.huge)
BodyOrientation.P = 100
BodyOrientation.D = 10
BodyOrientation.Parent = Part

local BodyForce = Instance.new("BodyForce",Part) -- *new*
BodyForce.Force = Vector3.new(0,workspace.Gravity * Part:GetMass(), 0) -- *new*

local function onRenderStep(deltaTime)
    local PlayerPosition = Character.HumanoidRootPart.Position
    local PlayerCFrame = Character.HumanoidRootPart.CFrame

    --BodyPosition.Position = PlayerPosition
    BodyOrientation.CFrame = PlayerCFrame

    --local OverallPosition = Vector3.new(PlayerPosition.x, PlayerPosition.y, PlayerPosition.z +3)

    local Distance = 10 -- *new*
    local OverallPosition = PlayerPosition - (PlayerCFrame.LookVector * Distance) -- *new*


    --Part.Position = OverallPosition
    -- If you are using BodyPosition, you don't need to adjust the part's position like this

    BodyPosition.Position = OverallPosition -- *new*

    Part.BodyGyro.CFrame = BodyOrientation.CFrame
end
RunService.RenderStepped:Connect(onRenderStep)

The BodyForce object I added negates the effects of gravity, allowing the part to be non-anchored.

This line keep the position of the part behind the player:

local OverallPosition = PlayerPosition - (PlayerCFrame.LookVector * Distance)

The BodyPosition instance sets the part's position to the position of the BodyPosition. You don't have to set the part's position and the BodyPosition's position, only the BodyPosition's position.

Dampeners are also a good idea for BodyGyro and BodyPosition, unless you want so freaky effects to occur ( I assume you don't ).

This line will change how far behind the player the part is positioned:

local Distance = 10 -- *new*

If you would like further assistance with your code, or would like something explained in a more detailed / different manner, feel free to contact me on Discord (phxntxsmic#2021)