As others have already said, the early Gray Death Legion books are set in exactly the same era as this game and have a similar type of feel with a fledgling mercenary unit putting itself together from scraps.
The Warrior trilogy by Michael A. Stackpole is basically the story of the Fourth Succession War, the next major conflict after the era the game is set in, and gets most of the long-term political storylines and generational character plots with members of the Inner Sphere "great houses" going.
The follow-on books written by Michael A. Stackpole carried the "main" storyline of the BT universe on through the Clan invasion (the Blood of Kerensky trilogy) and for a while after, until the main plot switched over to some of the "Twilight of the Clans" books (by Stackpole and a couple of other authors). Then Loren L. Coleman wrote the books carrying the main narrative of the Federated Commonwealth civil war up until the end of the series.
If you want a deeper character-driven story that takes place over a longer period of time than "Heir to the Dragon" and "Wolves on the Border" by Robert N. Charette are well-regarded, although the latter is focused on Wolf's Dragoons and their origin is tied in to the later Clan invasion storyline so it may be a little tougher to absorb for a newcomer to the setting.
If you want something with a focus on a small merc unit to feel more like the game, then besides the early Gray Death Legion books you might try Main Event by James D. Long. It's got some Solaris VII (the 'mech gladiator combat planet) action and the main character taking a tiny, newly-formed mercenary unit on a contract to go hold a backwater world against the Clans. It's a little cheeseball even by BT novel standards but I've always had a soft spot for it.
A couple to AVOID:
Far Country is very strange, unlike any other BT book really, in that it it actually has intelligent alien life in it. Most fans strongly dislike it as clashing with what the feel of the BT universe is suppose to be. If you treat it more like a novel or movie "in-universe" (like people in the 31st century went to see this as a cheesy popcorn flick) it's more tolerable.
The Sword and the Dagger is a very early novel which has a lot of technical stuff in it that contradicts canon established later on. It's also the one other BT book with maybe-aliens (although they're pretty incidental as opposed to a major part of the plot like in Far Country). This is partially because it was written by an author who had little knowledge of the setting when the original writer backed out and the contract had to be transferred at the last minute. It's not a bad book per se and its events do have some relevance as motivation for stuff that happens in the Warrior trilogy but it's highly skippable.