Although I am not an expert in this field (merely a beginning student), maybe I can help you out.
We will have to find a formula only using $\in$ and $=$ to describe $z = \langle \langle x, y \rangle, \langle u, v \rangle \rangle$. This essentially means we want to say that: '$z$ is an ordered pair of two ordered pairs $a$ and $b$. The ordered pair $a$ consists of the elements $x$ and $y$, the ordered pair $b$ consists of the elements $a$ and $b$.'
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To describe that $z$ is an ordered pair, you can use the formula:
$(\exists a \in z)(\exists b \in z)(\exists d \in a)(\exists f \in b) \left[ (\forall c \in z)(c = a \vee c = b) \wedge (\forall c \in a)(c = d) \wedge (d \in b) \wedge (\forall c \in b)(c = d \vee c = f) \right]$
Lets call this formula $o(z)$.
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Given that $z$ is an ordered pair, we can describe that $g$ is the first coördinate of $z$ with:
$(\forall a \in z)(g \in a)$
Lets call this formula $p(z, g)$.
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Given that $z$ is an ordered pair, we can describe that $i$ is the second coördinate of $z$ with:
$(\exists a \in z)(i \in a) \wedge (\forall b \in z)(\forall c \in z)(b \neq c \Rightarrow (i \notin b \vee i \notin c))$
Call this formula $q(z, i)$.
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By combining these formulas, we can describe $z = \langle \langle x,y \rangle, \langle u,v \rangle \rangle$ as follows:
$o(z) \wedge (\exists a)(\exists b)(o(a) \wedge o(b) \wedge p(z, a) \wedge q(z, b) \wedge p(a, x) \wedge q(a, y) \wedge p(b, u) \wedge p(b, v))$
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EDIT (NOT APPLICABLE ANYMORE)
As I do not have enough reputation to post a comment on other answers yet (:-s), I will comment on Asaf Karagila's answer in this way. The formula $\phi(u,v,z)$ you use is not correct, because $\phi(u,v,z)$ does not hold if and only if $z = \langle u, v \rangle$. It is for example possible that $z = \{ \{ u \} , \{ v \} \}$, or $z = \{ \{ u \} \}$ with $u \neq v$.
You really need to use the definition of ordered pairs with first and second coördinate I use in my answer, elsewise you won't have an 'if and only if':
$z = \langle u, v \rangle \mbox{ if and only if } o(z) \wedge p(z, u) \wedge q(z, v)$