Deep learning-based weather forecasting (DLWF) models leverage past weather observations to generate future forecasts, supporting a wide range of downstream applications, including tropical cyclone (TC) prediction. In this paper, we investigate their vulnerability to adversarial attacks, where subtle perturbations to the upstream forecasts can alter the downstream TC trajectory predictions. Although research into adversarial attacks on DLWF models has grown recently, it remains challenging to craft perturbed upstream forecasts that steer the downstream outputs toward attacker-specified trajectories. First, conventional TC detection systems are opaque, non-differentiable black boxes, making standard gradient-based attacks infeasible. Second, the extreme rarity of TC events leads to severe class imbalance problem, making it difficult to develop attack methods for perturbing upstream forecasts that produce realistic-looking cyclone paths aligned with attacker's target trajectories. To overcome these limitations, we propose Cyc-Attack, a novel method for perturbing the upstream forecasts of DLWF models to generate adversarial trajectories. The proposed method uses a differentiable surrogate model to approximate the TC detector's output, enabling the application of gradient-based attacks. Cyc-Attack also employs a skewness-aware loss function with kernel dilation strategy to address the imbalance problem. Finally, a distance-based gradient weighting scheme and regularization are used to constrain the perturbations and eliminate unrealistic-looking trajectories, thereby making the adversarial upstream forecasts less easily detectable. Our experiments show that Cyc-Attack achieves a higher true positive rate in matching the attacker's target trajectories, along with lower false alarm rates and stealthier perturbations than conventional attack methods.