From the classic version of this problem, we know $T_N$, the number of purchases required to collect N distinct coupons has $E(T_N) \approx NlogN$; $V(T_N) \approx N^2\pi^2 /6$.
Here I have got a problem asking me to give a least bound on $T_{20}$, such that with probability at least 0.95 I have collected all 20 coupons.
I am clueless with finding bounds on the random variable itself, since I could only write a Chebyshev Inequality on probability. Any help is appreciated!
Just for the record, the quantities you have given are approximations/bounds for the expectation and variance; they do not hold with equality. The Wikipedia page has the exact details and also gives a proper application of Chebychev's inequality.
However, if this is a homework question perhaps you are asked to suspend disbelief and assume the expectation and variance are exactly as stated.
Then applying Chebychev can be done as follows: $$P(T > E[T] + c) \le P(|T - E[T]| > c) \le \frac{V(T)}{c^2}.$$ Plug in your values of $E[T]$ and $V(T)$, and choose $c$ so that $V(T) / c^2 \le 0.05$. Then the probability that you need more than $E[T]+c$ purchases to get all coupons is $\le 0.05$; in other words, with $E[T]+c$ purchases you have collected all coupons with probability $\ge 0.95$.