Large language model (LLM) serving is becoming an increasingly important workload for cloud providers. Based on performance SLO requirements, LLM inference requests can be divided into (a) interactive requests that have tight SLOs in the order of seconds, and (b) batch requests that have relaxed SLO in the order of minutes to hours. These SLOs can degrade based on the arrival rates, multiplexing, and configuration parameters, thus necessitating the use of resource autoscaling on serving instances and their batch sizes. However, previous autoscalers for LLM serving do not consider request SLOs leading to unnecessary scaling and resource under-utilization. To address these limitations, we introduce Chiron, an autoscaler that uses the idea of hierarchical backpressure estimated using queue size, utilization, and SLOs. Our experiments show that Chiron achieves up to 90% higher SLO attainment and improves GPU efficiency by up to 70% compared to existing solutions.