Newtonian and Hamiltonian formulations are fully equivalent, and Newtonian mechanics does not introduce any additional difficulty in understanding chaos than Hamiltonian mechanics. Hamiltonian formalism does introduce much better understanding of reality, but none of that is directly related to chaos.
(Deterministic) Chaos arises when solving difference or differential equations that have nonlinear components, and a key feature of chaos is sensitive dependence on initial conditions. It matters not what the equations describe or where they came from. It can be a Hamiltonian system (e.g. solar system) or even a weather simplification (e.g. Lorenz equations). If the systems are nonlinear, chaos is very likely to arise. And, to study chaos you turn to tools unrelated to the origin of your differential equations, like e.g. Lyapunov exponents. Side comment: Poincare surfaces of section are also unrelated to the origin of your equations and can be applied just as well to weather systems.
By the way, there has been some comments (see @OON) that "you cannot have a Hamiltonian system with dissipation". Every Hamiltonian system where the Hamiltonian depends on time is equivalent to having a positive or negative dissipation, as one learns in introductory University physics (see any of the available textbooks on mechanics). For the experts: Notice that this can only apply to systems which respect the symplectic structure, since by definition a Hamiltonian system can be only symplectic. Of course, you can write aaaaany kind of model that is Newtonian but not symplectic. Have you ever seen such a thing in experiments though...?
EDIT: About Tong's lecture: (I have not watched these lectures) I am guessing that Tong is referring to the concept of the phase-space, which is not given special attention in Newtonian formalism (since, as @stafusa said in his answer, momentum and position are not treated on equal footing). Although the phase-space is not directly related to the existence of chaos (because stable systems also have a phase-space!) it does allow one to immediately understand whether a Hamiltonian system will be chaotic: in the Hamiltonian case the phase-space will be mixed or fully chaotic (ergodic). Of course, this requires some techniques of visualizing the phase-space, which is very easy with a computer.
An important side-note in the discovery of chaos: Chaos was first confronted by Poincare, when trying to answer the question "Is the solar system stable?", posed by King Oscar II of Sweden+Norway. At one point of his research, Poincare assumed that "a small change in the initial conditions will only have a small change in the evolved orbit". To his devastation, he realized later that this was not true. The concept of phase-space may have been useful for him to understand that. However, chaos was not truly appreciated until Lorenz came around 70 years later, and more specifically, computers came, which could solve unsolvable ODEs. Being able to solve these equations numerically was, at least for me, the crucial point in chaos theory history, and this does not depend on the origin of your equations.
