Towards a LotR campaign, part I

If you want to play a Middle-earth role-playing campaign set at the times of the Lord of the Rings, you'll be facing a simple dilemma: either you're a member of the fellowship of the ring, or you're out of the really relevant events.
From my point of view, the only solution is to really play the fellowship of the ring, but taking into account that not everything has to go just like in the book (otherwise, it's not a game). In particular, the composition of the fellowship might be significantly different still without breaking the overall Middle-earth flavour.
The main difficulties one would encounter in forming a fellowship of the ring are: the ringbearer, the hobbits in general, Gandalf, and Aragorn; all the other supporting characters are not that mandatory. We'll discuss them in order of increasing difficulty

Hobbits

Hobbits are great characters for the book, but not for the game. From my point of view, it is perfectly reasonable to just have a single hobbit in the party, the ringbearer.

Aragorn

The heir of Elendil, the future King of Men - not such an easy role.
Now, the consistency of the overall Middle-earth history requires that before the War of the Ring starts, the crown of Arnor is held by some sort of Aragorn, and Gondor is ruled by a steward.

If a character in the party wants to be heir of Elendil, he can very well play an Aragorn-like character, even if it will start at the same power level as the others (perhaps in this alternate timeline Aragorn is younger and didn't have time to grow as experienced as he's in the book); the precise details of the character don't have to match with the original one, even if flavour requires him to be somewhere around the fighter, ranger and warlord classes.

If nobody wants to be Aragorn, two options are left to the DM: either some Aragorn NPC exists, and move some of the events behind the scenes (e.g. brings an army in front of the Morannon to distract Sauron), or the two kingdoms of Arnor and Gondor will not be united at the end of the war - the Aragorn of this timeline will just play some minor role in the war, together with the rangers of the north, but won't gain the crown.

Gandalf

A real Gandalf PC is very hard to have, as he would be too powerful and too different from the other PCs; it might be ok only if the character is made just as powerful as the others - a difference from the book which still doesn't break the flavour too much.

Another option, probably easier, is just not to have Gandalf in the fellowship: perhaps he's still in the hands of Saruman, or was deeply wounded in the battle and can't take part to the quest; or perhaps he thinks his presence will be too hard to conceal, and would undermine the secrecy that is necessary for the fellowship to succeed.
In this case, Gandalf is likely to exist as an NPC and take care of some behind-the-scenes events, while the party if the PCs takes care of the Ring.
Having one spellcasting PC in the party is probably still useful (and perhaps even needed for the flavour), but he will be a less powerful one; in such an alternative timeline, the spellcaster might as well have been an apprentice of one of the Istari, or of Elrond.

The Ringbearer

The Ringbearer is of course necessary. And, unless you want to break the story quite a bit, it's very likely to be Frodo Baggins.
In order to make the game enjoyable for everyone, you have two options: Frodo as a PC, fighting in the front line just like in part of the book (e.g. at weathertop or in Balin's tomb), most likely rendered as a D&D rogue; a NPC Frodo that is handled collectively from the players, and mostly takes care of surviving.
The choice between the two is probably forced by whether you have a player that's willing to take a Hobbit character and the burden of Isildur's Bane, or not.