How to define a weighted cone? Let $F : I \to C$ be a diagram in $C$, and $N$ an object of $C$. A cone from $N$ to $F$ is a family of morphism $P_X : N \to F(X)$ such that for every morphism $f : i1 \to i2$ in $I$, $F(f) \circ P_X = P_Y$. And the limit of $F$ is the universal cone.
If $C$ is now a $Cat$-enriched category and $W : I \to Cat$ is given as a weight, is there a notion of $W$-weighted cone such that a $W$-weighted limit is a universal $W$-weighted cone? If yes, how do you define such a $W$-weighted cone? And what does it mean for it to be universal? 

Recall that a cone from $N$ to $F$ is a natural transformation from the constant functor $\Delta N$ to $F$. A cone is universal if it represents the functor $[I,C](\Delta -,F) : C^\text{op} \to \textbf{Set}$, i.e. if it induces isomorphisms $$C(X,N) \cong [I,C](\Delta X,F)$$ for all objects $X$ of $C$. Note that there is a natural isomorphism $$[I,C](\Delta X,F) \cong [I,\textbf{Set}](\Delta 1, C(X,F-)),$$ so we can see ordinary (universal) cones as $\textbf{Set}$-enriched $\Delta 1$-weighted (universal) cones.
Now in the 2-categorical context, let $\hat{F}$ denote the 2-functor $C^\text{op} \to [I,\textbf{Cat}]$ with $\hat{F}X = C(X,F-)$.
A $W$-weighted cone from $N$ to $F$ is then a 2-natural transformation $W \to \hat{F}N$, and such a cone is universal if it represents the 2-functor $[I,\textbf{Cat}](W,\hat{F}-) : C^\text{op} \to \textbf{Cat}$, i.e. if it induces isomorphisms of categories $$C(X,N) \cong [I,\textbf{Cat}](W,C(X,F-))$$ for all objects $X$ of $C$. This defines the universal property of the $W$-weighted limit of $F$.
Re your comment: 
A 2-natural transformation $\theta : F\to G : \mathcal{A} \to \mathcal{B}$ is given by a family of 1-morphisms $\theta_A : FA \to GA$ of $\mathcal{B}$, indexed by the objects A of $\mathcal{A}$, such that the diagram of functors
$$
\require{AMScd}
\begin{CD}
\mathcal{A}(A,B) @>F>> \mathcal{B}(FA,FB)\\
@VGVV @VV\mathcal{B}(1,\theta_B)V\\
\mathcal{B}(GA,GB) @>>\mathcal{B}(\theta_A,1)> \mathcal{B}(FA,GB)
\end{CD}$$
commutes for all objects $A,B$ of $\mathcal{A}$. This amounts to the usual naturality condition, plus an extra condition that for every 2-cell $\alpha : f \to g : A \to B$ of $\mathcal{A}$, we have $$\theta_B \circ F\alpha = G\alpha \circ \theta_A,$$ where $\circ$ denotes horizontal composition (''whiskering'' in this case) in $\mathcal{B}$.
Weighted limits and enriched natural transformations are elementary definitions of enriched category theory, which may be found in the standard reference on the subject, Kelly's book Basic concepts of enriched category theory.
