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 terms 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 methods and LLM-based zero-shot prompting methods on two public datasets.