Can we proxy-re-encrypt using homomorphic encryption schemes? Homomorphic encryption schemes are PKE schemes with an additional special method Evaluate. 

The Evaluate method takes input any function (as boolean circuit) and encrypted inputs of the function and evaluates the circuit. The results of such function are still encrypted under the public key $PK$.

Now, if the user needs to re-encrypt the data with a new public key $PK^1$, the user should decrypt the data with older private key $SK$ first, and then encrypt again with new $PK^1$. 

Or how can I efficiently proxy-re-encrypt using FHE schemes without decrypting the data?

Note that if the user runs the re-encrypt circuit as input to the Evaluate method with $PK^1$ as parameter, the result would still  be encryption of (encryption of original data with new $PK_1$) with the older $PK$.
YES. It is pointed out by Gentry (in his thesis) when he consider bootstrapping and re-encryption.

Let us denote the old key pair by $(PK, SK)$ and the new key pair by $(PK^1, SK^1)$.

What the user wants is to re-encrypt an old ciphertext $C = \mathsf{Enc}(PK, M)$ into a new ciphertext $C^1$ which is decrypted into $M = \mathsf{Dec}(SK^1,C^1)$.

To do so without direct decryption, we virtually decrypt $C$ in the draft chamber guarded by $PK^1$, that is, we evaluate a decryption circuit with $PK^1$.
Correctly speaking, we do the following:


Generate $\widetilde{SK}_i = \mathsf{Enc}(PK^1,SK_i)$, where $SK_i$ denotes the i-th bit of $SK$.
Compute $\widetilde{C}_i = \mathsf{Enc}(PK^1,C_i)$, where $C_i$ denotes the i-th bit of $C$.
Evaluate the decryption circuit. That is, $C^1 = \mathsf{Eval}(PK^1, \mathsf{Dec}, (\widetilde{SK}_1,\dots,\widetilde{SK}_n,\widetilde{C}_1,\dots,\widetilde{C}_m))$.


From the correctness of evaluation, we have
$$
\begin{align}
\mathsf{Dec}(SK^1,C^1)
 &amp;= \mathsf{Dec}(\mathsf{Dec}(SK^1,\widetilde{SK}_1),\dots,\mathsf{Dec}(SK^1,\widetilde{SK}_n),\mathsf{Dec}(SK^1,\widetilde{C}_1),\dots,\mathsf{Dec}(SK^1,\widetilde{C}_m))\\
 &amp;= \mathsf{Dec}(SK_1,\dots,SK_n,C_1,\dots,C_m) \\
 &amp;= \mathsf{Dec}(SK,C)\\
 &amp;= M
\end{align}
$$
as we want.

This is just re-encryption Gentry suggested in his thesis.