Answer by taking a quote from the following article:

David Thompson was born in Westminster, Middlesex, to recent Welsh migrants David and Ann Thompson. When Thompson was two, his father died. Due to the financial hardship with his mother without resources, Thompson and his older brother were placed in the Grey Coat Hospital, a school for the disadvantaged of Westminster. Thompson graduated to the Grey Coat mathematical school, where he was introduced to basic navigation skills.

Thompson arrived in Churchill (now in Manitoba) and was put to work as a secretary, copying the personal papers of the governor of Fort Churchill, Samuel Hearne. The next year he was transferred to nearby York Factory, and over the next few years spent time as a secretary at Cumberland House, Saskatchewan, and South Branch House before arriving at Manchester House in 1787. During those years he learned to keep accounts and other records, calculate values of furs (It was noted that he also had several expensive beaver pelts at that time even when a secretary's job would not pay terribly well), track supplies and other duties.  On 23 December 1788, Thompson seriously fractured his leg, forcing him to spend the next two winters at Cumberland House convalescing. It was during this time that he greatly refined and expanded his mathematical, astronomical, and surveying skills under the tutelage of Hudson's Bay Company surveyor Philip Turnor. It was also during this time that he lost sight in his right eye.  In 1790, with his apprenticeship nearing its end, Thompson requested a set of surveying tools in place of the typical parting gift of fine clothes offered by the company to those completing their indenture. He received both. He entered the employ of the Hudson's Bay Company as a fur trader. In 1792 he completed his first significant survey, mapping a route to Lake Athabasca (where today's Alberta/Saskatchewan border is located). In recognition of his map-making skills, the company promoted Thompson to surveyor in 1794. He continued working for the Hudson's Bay Company until 23 May 1797 when, frustrated with the Hudson's Bay Company's policies over promoting the use of alcohol with indigenous people in the fur trade, he left. He walked 130 kilometres (80 mi) in the snow in order to enter the employ of the competition, the North West Company. There he continued to work as a fur trader and surveyor.

What areas did he map with HBC?
mapping a route to Lake Athabasca