If $L$ is a regular language, how to prove $L_1 = \{ uv \mid u \in L, |v| =2 \}$ is also regular?
Your idea is correct.

You should take the following path to formally write it up/prove it. First assume that $L$ is accepted by some DEA $M=(Q,\Sigma,\delta,q_0,F)$ then define a new NEA $M'=(Q',\Sigma,\delta',q'_0,F')$ based on $M$. Just express you ideas formally, for example start with 


$Q'=Q\cup\{q_\text{new1},q_\text{new2}\}$, 
$F'=\{q_\text{new2}\}$,
....


Then you should prove, using the acceptance criteria for $M$ and $M'$, that $w\in L(M) \iff wv\in L(M')$, for all $v\in \Sigma^2$.

Alternative approach

The following idea will also prove the statement but results in an easier proof. First show that the language of words with exactly two characters is regular, and then argue with the closure properties for regular languages.