For a very elementary example let $Y$ be an uncountable set, $y_0$ and $y_1$ distinct points of $y$, and $p$ a point not in $Y$. Let $X=Y\cup\{p\}$, $\mathscr{U}=\wp(Y\setminus\{y_0,y_1\})$, and $\mathscr{V}=\big\{U\cup\{y_0,y_1\}:U\in\mathscr{U}\big\}$. Finally, let
$$\tau=\mathscr{U}\cup\mathscr{V}\cup\{X\setminus C:C\in\mathscr{U}\cup\mathscr{V}\text{ and }C\text{ is countable}\}\;;$$
then $\tau$ is a topology on $X$. Every member of $\tau$ contains either both or neither of the points $y_0$ and $y_1$, so $\langle X,\tau\rangle$ is not $T_0$. $Y$ is sequentially closed, since the convergent sequences in $Y$ are those that are eventually constant or eventually in the set $\{y_0,y_1\}$, but $p\in\operatorname{cl}_XY$, so $Y$ is not closed in $X$.
One that fails to be sequential but still has non-trivial convergent sequences can be obtained by letting $Y=\omega_1+1$ with the order topology $\tau'$ and $p$ be a point not in $Y$. Let $X=Y\cup\{p\}$, and let
$$\tau=\{U\in\tau':0\notin U\}\cup\big\{U\cup\{p\}:0\in U\in\tau'\big\}\;$$
then $\langle X,\tau\rangle$ is not $T_0$, because every open set contains either both or neither of the points $0$ and $p$, and it’s not sequential, because $X\setminus\{\omega_1\}$ is a sequentially closed set that is not closed.
Each of these is the result of applying a very simple ‘machine’ to an arbitrary non-sequential space $\langle Y,\tau\rangle$. Fix some $y_0\in Y$, let $Z=\{0,1\}$ have the indiscete topology, and let $X=(Y\times Z)/{\sim}$, where $\langle y_1,z_0\rangle\sim\langle y_2,z_1\rangle$ if and only if either $\langle y_1,z_0\rangle=\langle y_2,z_1\rangle$, or $y_1=y_2\ne y_0$. This simply doubles the point $y_0$, turning it into two distinct points with identical nbhds in $X$, in such a way that $Y$ is the Kolmogorov quotient of $X$.
In fact it is true in general that if $X$ is non-sequential, then the Kolmogorov quotient $Y$ of $X$ is non-sequential, so every example of a non-sequential, non-$T_0$ space can be obtained by ‘fattening up’ a $T_0$ non-sequential space, albeit not necessarily at just one point.
