How to show a sequence of measure converges weakly?

The doubt I have is that in all equivalent definitions of weak convergence of finite measures via the Portmanteau theorems, some knowledge of $\mu$ is required to check those conditions, for example to one should know about the continuity sets of the $\mu$, in order to use the condition, $\mu_n(A) \to \mu{}(A), \forall A \ni \mu(\partial A)=0$.

What are the methods to show weak convergence(or non convergence) of a sequence of finite measures, when you have no knowledge of what the limiting measure would be?

For eg, how to show the uniform measure on {1,$\cdots$,n} does not converge to any probability measure.

One consequence of weak convergence is tightness: If $\mu_n$ converges weakly then , given $\epsilon >0$ there exists a compact set $K$ such that $\mu_n(K^{c}) <\epsilon$ for all $n$. So if $(\mu_n)$ is not tight then it is not convergent. This is often used to show that a given sequence does not converge. This argument works in your example.