Generalized autoregressive conditionally heteroskedastic (GARCH) processes are widely used for modelling financial returns, with their extremal properties being of interest for market risk management. For GARCH(p,q) processes with max(p,q) = 1 all extremal features have been fully characterised, but when max(p,q) is greater than or equal to 2 much remains to be found. Previous research has identified that both marginal and dependence extremal features of strictly stationary GARCH(p,q) processes are determined by a multivariate regular variation property and tail processes. Currently there are no reliable methods for evaluating these characterisations, or even assessing the stationarity, for the classes of GARCH(p,q) processes that are used in practice, i.e., with unbounded and asymmetric innovations. By developing a mixture of new limit theory and particle filtering algorithms for fixed point distributions we produce novel and robust evaluation methods for all extremal features for all GARCH(p,q) processes, including ARCH and IGARCH processes. We investigate our methods' performance when evaluating the marginal tail index, the extremogram and the extremal index, the latter two being measures of temporal dependence.