Architected Cellular Materials (ACM) are increasingly used in modern engineering applications due to their specific stiffness and strength, and also their versatility in application cases. Metallic ACMs can be fabricated mainly via additive manufacturing processes such as Selective Laser Melting. ACMs are grouped into different classes based on their mechanical behaviour and/or geometrical characteristics.Amongst the different classes of ACMs available in literature, this work will focus on the design of strut-based lattice structures. This class of ACMs includes some well-known topologies, such as cubic, kelvin, diamond and octet. The design strategy of the lattice structures presented in this work is articulated in two steps. Firstly, a wide campaign of thermomechanical homogenisation analyses is conducted on the representative volume element (RVE) of the strut-based lattice structure, for each topology belonging to this class. The goal is to determine the effective thermomechanical properties of the lattice structure at the upper scale, where it is modelled as an equivalent homogeneous anisotropic material. Specifically, the thermal conductivity tensor, the elasticity tensor, the limit stresses and the tensor of coefficients of thermal expansion at the upper scale are determined as a function of the relative density of the RVE in order to formulate physically meaningful penalty schemes to be used during the topology optimisation (TO) phase. Secondly, the design of the graded lattice structure is performed by using a special density-based TO method, wherein the pseudo density field is represented through non-uniform rational basis spline (NURBS) entities. The main features of the proposed approach are: 1) the continuity of the pseudo-density field, which, together with the local support property of NURBS entities, does not require the introduction of artificial filtering schemes to reduce the mesh dependency of the solution and avoid/reduce the well-known checkerboard effect, and 2) the CAD compatibility of the solution. The proposed design strategy will be validated on 2D and 3D benchmark problems taken from literature.