Is there a description of finite groups whose all quotients have trivial center? Is it true that only direct products of non-abelian simple groups have this property?
The answer to your second question is negative. Take a finite simple group $G$ and its nontrivial irreducible representation $V$ over finite field. Then the semi-direct product of $G$ and $V$ has a unique nontrivial quotient, the group $G$ itself.
The quotients of a finite group, G/N have minimal normal subgroups K/N. These are called chief factors. Chief factors are divided into two kinds, central chief factors and eccentric chief factors. The central ones are precisely those such that K/N ≤ Z(G/N). You are therefore asking for a classification of groups all of whose chief factors are eccentric.
Another way to say this that sounds clever is that G has no chief factor of prime order.
Such a group cannot have any central "top" factors, and in particular G must be a perfect group. Perfect groups are not entirely easy to classify, but one standard view of them is to build them layer by layer, and under this view you simply never adjoin a central factor.
However, such a classification may not be very useful in concrete circumstances. The things you can adjoin are any (fixed point free, that is, noncentral) G-module repeatedly to construct the solvable radical, and any wreath product of G with a non-abelian simple group.
For every permutation representation of A5, you can take A5 wr A5 to get such examples. For any A5-module V (with no central A5-composition factors), you can take the semi-direct product of A5 with V.
There are 26 such groups (not counting G=1) of order at most 10,000; most are just the simple groups of those orders. There are also 2^4:A5, 4^2:A5, 2^3:L3(2), 2^3.L3(2), A5 x A5, 3^4:A5, 3^4.A5, 2^4:A6, 5^3:A5, 5^3.A5. The first wreath product example is A5 wr A5 of order 60^6.