Spelling out explicitly the data of a two step filtration in terms of pieces and gluing data Let $V$ be an object of some stable infinity category (nothing is lost by taking spectra but I see no reason to state the question in this way as it is irrelevant) and suppose we have a two step filtration:
$$V_0 \subset V_1 \subset V_2$$
Lets denote $A=V_0$, $B=V_1/V_0$, $C= V_2/V_1$, $D=V_1$, $E=V_2/V_0$, $F=V_2$.
The data of the filtration gives us two different ways to build $F$ by iterated extensions. 
First option:
$$D \longrightarrow B \overset{\alpha}{\longrightarrow} A[1]$$
$$F \longrightarrow C \overset{\gamma}{\longrightarrow} D[1]$$
Second option:
$$E \longrightarrow C \overset{\beta}{\longrightarrow} B[1]$$
$$F \longrightarrow E \overset{\delta}{\longrightarrow} A[1]$$
Of course there must be relations between $\{\alpha,\beta,\gamma,\delta\}$ since they came from a single 2-step filtration.

Question: Is there a general set of relations (expressed via cup product massey products and perhaps even some extra coherence data) one can define on collections of the form $\{\alpha,\beta,\gamma,\delta\}$ which will give a bijective correspondence between 2-step filtered objects with subquotients $A,B,C,D,E,F$ and collections of the form $\{\alpha,\beta,\gamma,\delta\}$ which satisfy these relations?

 A: Technically speaking the answer to your question is no, in the sense that the data of $(\alpha,\beta,\gamma,\delta)$ alone does not determine the filtered object $V_0 \subseteq V_1 \subseteq V_2$. However, a variant on your construction does have a positive answer, and this is closely related to Massey products. Recall first that if $(A_\bullet,d)$ is a (homologically graded) dg-algebra then an $n$-fold Massey system on $A$ is a collection of elements $a_{i,j} \in A_{-1}$ for $0 \leq i < j \leq n$ with $(i,j) \neq (0,n)$ satisfying the relation $\partial a_{i,j} = \sum_{k=i+1}^{j-1} a_{i,k}a_{k,j}$ (in particular, $\partial a_{i,i+1} =0$). In this case, the element $b := \sum_{k=1}^{n-1}a_{0,k}a_{k,n} \in A_{-2}$ is closed and its class $[b] \in H_{-2}(A)$ is called the value of the Massey system $\{a_{i,j}\}$. Classically this value is often considered as the $n$-fold Massey product of the classes $[a_{i,i+1}] \in H_{-1}(A)$, although one should remember that this value depends on the choice of a Massey system, and not just on the $[a_{i,i+1}]$. A trivialization of a Massey system $\{a_{i,j}\}$ is an element $a_{0,n} \in A_{-1}$ satisfying $\partial a_{0,n} = \sum_{k=1}^{n-1}a_{0,k}a_{k,n}$, i.e., witnessing the vanishing of the associated $n$-fold Massey product.
The main observation (which I believe, under a somewhat different formulation, is essentially due to Dwyer), is that the data $(\{a_{i,j}\},a_{0,n})$ of an $n$-fold Massey system equipped with a trivialization classifies (right) $A$-modules $M$ equipped with an ascending filtration 
$$0 = M_{-1} \to M_0 \to M_1 \to ...\to M_n=M $$ 
such that each homotopy cofiber $M_{i}/M_{i-1}$ is (identified with) $A$. For example, suppose that $n=1$. Then the data of a trivialized Massey system is just the data of a closed element $a_{0,1} \in A_{-1}$, which can be considered as a map of $A$-modules $a_{0,1}: A \to A[1]$. One then defines $M$ to be the homotopy fiber of this map, so that $M$ sits in a homotpy fiber sequence $A \to M \to A$. In particular, $M$ is equipped with a $2$-step filtration 
$$ 0 \to A \to M $$ 
whose successive cofibers are $A$. Now consider the case $n=2$. Then we are given closed elements $a_{0,1},a_{1,2} \in A_{-1}$ and a trivialization $a_{0,2} \in A_{-1}$ satisfying $\partial a_{0,2} = a_{0,1}a_{1,2}$. We may then consider $a_{0,1}$ and $a_{1,2}$ as $A$-module maps $a_{0,1}: A[1] \to A[2]$ and $a_{1,2}: A[0] \to A[1]$, and consider $a_{0,2}$ as a null-homotopy of their composition $a_{0,1} \circ a_{1,2}$. Let $M_{0,1}$ and $M_{1,2}$ be the homotopy fibers of $a_{0,1}$ and $a_{1,2}$ respectively. Then the null-homotopy $a_{0,2}$ determines a lift of $a_{1,2}: A \to A[1]$ to a map $\tilde{a}_{1,2}: A \to M_{0,1}$. Let $M_{0,2}$ be the homotopy fiber of $\tilde{a}_{1,2}$. Then $M_{0,2}$ is equipped with a $3$-step filtration
$$ 0 \to A \to M_{0,1}[-1] \to M_{0,2} $$ 
whose successive cofibers are $A$.
This idea can be generalized to describe filtered $A$-modules with other types of successive cofibers. Indeed, if $B_0,...,B_n$ are fixed $A$-modules, then we can define the notion of a Massey system on $B_0,...,B_n$ to be a collection of elements $a_{i,j} \in \underline{\rm Hom}_{-1}(B_j,B_i)$ (where $\underline{\rm Hom}_\bullet$ means the mapping chain-complex of $A$-modules) for $(i,j) \neq (0,n)$ such that $\partial a_{i,j} = \sum_{k=i+1}^{j-1} a_{i,k}\circ a_{k,j}$, and define a trivialization of $\{a_{i,j}\}$ to be an element $a_{0,n} \in \underline{\rm Hom}_{-1}(B_n,B_0)$ such that $\partial a_{0,n} = \sum_{k=1}^{n-1}a_{0,k} \circ a_{k,n}$. Then the data of a Massey system on $B_0,...,B_n$ equipped with a trivialization then classifies $A$-modules equipped with an $(n+1)$-step filtration whose successive cofibers are $B_0,...,B_n$. Let's call such objects $(B_0,...,B_n)$-filtered $A$-modules. You can also ask, what about Massey systems $\{a_{i,j}\}$ not-necessarily trivialized? Well, these will classify triples $(M_{0,n-1},M_{1,n},f)$ where $M_{0,n-1}$ is a $(B_0,...,B_{n-1})$-filtered $A$-module, $M_{1,n}$ is a $(B_1,...,B_n)$-filtered $A$-module and $f$ is an equivalence of $(B_1,...,B_{n-1})$-filtered $A$-modules between ${\rm cofib}(B_0 \to M_{0,n-1})$ and ${\rm fib}(M_{1,n}\to B_n)$. The associated $n$-fold Massey product $[\sum_{k=1}^{n-1}a_{0,k} \circ a_{k,n}] \in H_{-2}(\underline{\rm Hom}(B_n,B_0))$ (which can be considered as a homotopy class of maps $B_n \to B_0[-2]$) is then the obstruction to finding a $(B_0,...,B_n)$-filtered $A$-module $M_{0,n}$ such that ${\rm fib}(M_{0,n}\to B_n) \simeq M_{0,n-1}$ and ${\rm cofib}(B_0 \to M_{0,n}) \simeq M_{1,n}$.
Of course, what can be done for modules can usually be done for an arbitrary stable $\infty$-category. Indeed, if $B_0,B_1,...,B_n$ are fixed objects in a stable $\infty$-category ${\cal C}$ then we can classify $(B_0,B_1,...,B_n)$-filtered objects by suitable Massey systems $\{a_{i,j}\}$, where the $a_{i,i+1}$ are maps of the form $a_{i,i+1}: B_{i+1}[n-i-1] \to B_{i}[n-i]$, $a_{i,i+2}$ is a null-homotopy of the composition $a_{i,i+1} \circ a_{i+1,i+2}$, etc. The case you are asking about is the case of a $3$-step filtration with successive cofibers $A,B,C \in {\cal C}$. In this case $(A,B,C)$-filtered objects are classified by the data $(\alpha',\beta,h)$ consisting of maps $\alpha': B[1] \to A[2]$ and $\beta: C \to B[1]$ and a null-homotopy $h$ of $\alpha' \circ \beta$. Here $\alpha'$ is a shift of your $\alpha$. The null-homotopy $h$ is actually determined by either a lift of $\beta$ to $\gamma: C \to D[1]:= {\rm fib}(\alpha')$ or an extension of $\alpha'$ to $\delta': E[1] := {\rm cofib}(\beta) \to A[2]$. This is essentially what you did with your $\gamma$ and $\delta$. However, to get the null-homotopy $h$ it is not sufficient to remember just $\gamma$ (or $\delta$), one also need to keep the homotopy relating the composition $C \stackrel{\gamma}{\to} D[1] \to B[1]$ to $\beta$. Of course if you do this for both $\gamma$ and $\delta$ you end up encoding this null-homotopy twice, so you will need to add a homotopy relating the two choices. Alternatively, just drop one of $\gamma,\delta$ (and add the missing homotopy instead).
A: Most of this is just the octahedral axiom for triangulated categories.  It would be painful to typeset it in MathJax, but the axiom gives you an octahedral diagram in which four triangular faces are distinguished triangles and the other four faces are commutative.  Four of the edges are your $\alpha,\dotsc,\delta$, and the others are just inclusions $V_i\to V_j$ or projections $V_j\to V_j/V_i$ or the obvious maps $V_1/V_0\to V_2/V_0\to V_2/V_1$.
You might want more than what the octahedral axiom gives you, but even if that is the case, I think that you should explain what you want with reference to the octahedral axiom.
