I can't speak on what things were like at the upper levels of command, however, Richard Holmes' book Sahib on the Indian army of the 19th and 18th centuries does talk about this a little, so here's what little I remember.
When an Army regiment was brought to serve with the Company, a HMs (for his/her majesty's) was prefixed to the title. For example, the 28th Regiment of Foot would become HMs 28th Regiment of Foot, to avoid confusion with the Company's separate numbering system for regiments. After this status was granted it was treated by the company like any other of its Regiments.
HMs regiments often took higher casualties during campaign compared to Company regiments, though this can be attributed to several things. Lack of familiarity with the local flaura and fauna, and lack of resistance to disease which many older members of the company's army had gained, rather than just Company commanders being happier to spend HMs Regiments lives, as many older Company commanders had a history in the British Army (The Duke of Wellington served in the Company's army early in his career, and you can find many Waterloo veterans on the company's list up to the Mutiny).
Relations between Army and Company officers could be cold, for two reasons. Firstly, Army officers were automatically one rank above their own in seniority when serving in India, so an Army Captain outranked a Company one, even though they both held the same rank. This is the same as is found between Guard Regiment officers and ones in the the rest of the army, and implies an inherent superiority. Conversely, Company officers were better paid, and had greater opportunity to make extra money on campaign, through a system of extra pay granted while in action, known as "Batta", which was granted to all ranks, though not equally. So members of both Armies had reasons to dislike each other.
Edit: Apologies for not mentioning native Indian soldiers (who of course made up the grand majority of the Company's army) in any respect, but as the majority of official interaction between Army and Company happened between white officers (this is less true in the 18th Century, however, when relations between British upper class in India and Indian aristocracy were quite warm), there isn't much I can say regarding them in relation to this topic.
In any period in particular? The Company was the longest lived joint stock company, so this could cover more than 2.5 ish centuries.
Edit: this source says a little bit about it and that it was "the oldest".
What are some good books to learn about the history of the East India Company?