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We have constructed a spherically symmetric structure model within an otherwise FRW universe filled with perfect fluid and studied its quasi-local characteristics. This is done by using the Lema\^{i}tre solution of the Einstein equations and by suggesting an algorithm to integrate it numerically. The result shows how a dynamical black hole evolves within the FRW universe, its decoupling from the expanding parts of the model, the structure of its space-like apparent horizon, the limiting case of the dynamical horizon tending to a slowly evolving horizon, and the decreasing mass in-fall to the black hole. The quasi-local features of this cosmological black hole may not be inferred from the Newtonian approximation although the gravity outside the structure is very weak.
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