Hierarchical text classification aims to categorize each document into a set of classes in a label taxonomy. Most earlier works focus on fully or semi-supervised methods that require a large amount of human annotated data which is costly and time-consuming to acquire. To alleviate human efforts, in this paper, we work on hierarchical text classification with the minimal amount of supervision: using the sole class name of each node as the only supervision. Recently, large language models (LLM) show competitive performance on various tasks through zero-shot prompting, but this method performs poorly in the hierarchical setting, because it is ineffective to include the large and structured label space in a prompt. On the other hand, previous weakly-supervised hierarchical text classification methods only utilize the raw taxonomy skeleton and ignore the rich information hidden in the text corpus that can serve as additional class-indicative features. To tackle the above challenges, we propose TELEClass, Taxonomy Enrichment and LLM-Enhanced weakly-supervised hierarchical text classification, which (1) automatically enriches the label taxonomy with class-indicative topical terms mined from the corpus to facilitate classifier training and (2) utilizes LLMs for both data annotation and creation tailored for the hierarchical label space. Experiments show that TELEClass can outperform previous weakly-supervised hierarchical text classification methods and LLM-based zero-shot prompting methods on two public datasets.