Constructing a Hamiltonian for $N$-qubits Let us assume we have a qubit with an internal Hamiltonian $H_0 = \sum_i \varepsilon_i |i\rangle\langle i|$.
Now let's assume we have 2 such qubits. How would their joint Hamiltonian look like?
I have here a formula for N-qubits which I don't fully understand.
\begin{equation} 
\begin{split}
H_0^{N} = \sum_k |1\rangle_k\langle 1|_k \otimes_{j \neq k} \mathbb{I}^j
\end{split}
\end{equation}
So for N=2 I'd get
\begin{equation} 
\begin{split}
H_0^{2} =  |1\rangle_1\langle 1|_1 \otimes_{2} \mathbb{I}^2 + 
|1\rangle_2\langle 1|_2 \otimes_{1} \mathbb{I}^1
\end{split}
\end{equation}
Which I don't understand. 
 A: Any Hamiltonian can be written in the form you give
$$H=\sum_i\varepsilon_i|i\rangle\langle i|$$
as long as the eigenstates form a basis. This is still true for a many-body system. So for a single qubit you'd have
$$
H_1=\varepsilon_0|0\rangle\langle 0|+\varepsilon_1|1\rangle\langle 1|
$$
And for two it'd look like
$$
H_2=\varepsilon_{00}|00\rangle\langle 00|+\varepsilon_{01}|01\rangle\langle 01|+\varepsilon_{10}|10\rangle\langle 10|+\varepsilon_{11}|11\rangle\langle 11|
$$
This representation is completely general and so offers no real insight into anything. If you assume all of your qubits are governed by the same Hamiltonian and independent of the others (noninteracting) then you can write the joint Hamiltonian as a sum over each individual one like you have above. This is generally the situation of interest.
$$
H=^{(*)}\sum_j H_j=\sum_j\sum_i\varepsilon_i|i\rangle_j\langle i|_j
$$
*Here $H_j$ means $H_j\otimes_{k\neq j} \mathbb{I}_k$ and $|i\rangle_j\langle i|_j=|i\rangle_j\langle i|_j\otimes_{k\neq j} \mathbb{I}_k$ i.e. they just act on qubit $j$
So for a two qubit system you'd have, fully written out:
$$
H_2=\varepsilon_0|0\rangle\langle 0|\otimes\mathbb{I}+\varepsilon_1|1\rangle\langle 1|\otimes\mathbb{I}+\varepsilon_0\mathbb{I}\otimes|0\rangle\langle 0|+\varepsilon_1\mathbb{I}\otimes|1\rangle\langle 1|
$$
Or to more closely match your notation (which is not useful when written out in full, I'd only use it to compactify expressions)
$$
H_2=\varepsilon_0|0\rangle_1\langle 0|_1\otimes_2\mathbb{I}_2+\varepsilon_1|1\rangle_1\langle 1|_1\otimes_2\mathbb{I}_2+\varepsilon_0|0\rangle_2\langle 0|_2\otimes_1\mathbb{I}_1+\varepsilon_1|1\rangle_2\langle 1|_2\otimes_1\mathbb{I}_1
$$
