Since I don't know whether you want to use TM's and because I'm unsure what TM's you can get without going through too much trouble, this rate will mostly feature moves that can be learned by HM / Level-up, so they may not be the best, but in-game, that doesn't really matter that much. If you want I could rate this team and include egg-moves, TM's and the like in the rate, but for now, I won't.
Charizard
Lonely / Naughty
~ Flamethrower
~ Fly
~ Metal Claw
~ Slash
Charizard is not amongst the fastest of Pokémon, but still has a decent base speed of 100. Enough speed to sacrifice the +speed nature for a +attack one, to boost both Metal Claw and Slash. I realise that this is your starter, and that you won't be able to change the nature. It is, however, not listed in the OP, so I'll give you one anyway.
Marowak @ Thick Club
Adamant @ Rock Head
~ Bonemerang
~ Double Edge
~ False Swipe
~ Rock Slide
All you really need is Bonemerang+Double Edge. The only Pokémon that resist this combination are Shedinja, Aerodactyl and Skarmory. If you want to teach Marowak Rock Slide, that would solve these problems, so I'll list it as an option. False Swipe is just a filler, but helps with catching stuff. Adamant to boost attack, the same goes for Thick Club. Rock Head so that you negate the recoil damage from Double Edge.
Raichu
Mild / Rash
~ Thunderbolt
~ Thunder Wave
~ Slam
~ Quick Attack / Light Screen
The Pikachu line has got an awful movepool, so this is all it can do with the level-up moves it gets. Thunderbolt gets STAB and is therefore your main attack. Thunder Wave to both slow things down and possibly render them unable to attack. Slam for some coverage. The last move is mostly a filler. Quick Attack gets priority, so you can use it to pick of weakened foes. Light Screen is more of a supporting move. Mild/Rash boosts Special Attack, so that Thunderbolt is as powerful as it gets.
I wouldn't run a Pikachu with Light Ball, since that only boosts Special Attack. Raichu doesn't get quite as much Special Attack as a Pikachu with Light Ball, but has higher Speed and Attack, and the Special Attack difference isn't all that big.
Exeggutor
Sassy
~ Leech Seed
~ Hypnosis / Poisonpowder
~ Solarbeam
~ Uproar
Another Pokémon with a barren movepool. Sassy lowers speed, but boosts Special Defence. Combined with Leech Seed, and maybe a Light Screen, this allows for stalling mostly. Poisonpowder helps with that even more, Hypnosis puts stuff to sleep, rendering them unable to attack you. Solarbeam is the only offensive Grass move it gets, Uproar for some sort of coverage.
Lapras
Quiet @ Water Absorb
~ Surf
~ Ice Beam
~ Body Slam
~ Confuse Ray
Surf and Ice Beam are two great STAB moves. Body Slam gets coverage against other Water types and has a nifty chance of paralyzing the opponent. Confuse Ray acts as a filler, but might help. Don't rely on it too much though, it'll probably fail you when you need it most. Quiet nature boosts Special Attack in exchance for some speed loss.
Now, for the last Pokémon I'd personally ditch Dragonite. It only adds further to your Rock and Ice Weakness. As for what you could use: a Fighting type covers Rock and hits both Rock and Ice for SE damage. That being said, you really don't need much to beat in-game, and if you really want a great team, you should probably be using some different Pokémon. But that's not always what it's about. So I'll give you a set for Dragonite, and I'll look for a Fighting type Pokémon you could use.
Dragonite
Mild / Rash
~ Agility
~ Outrage
~ Surf
~ Fly
It is a shame to lower Dragonite's defences, but it's really all you can do if you want to run a set that doesn't require breeding and TM-moves. Outrage+Surf is unresisted, and both run off the Special Attack stat, hence the Special Attack boosting natures. Fly is a filler, but a STAB move nonetheless.
Hitmonlee
Jolly
~ Hi Jump Kick
~ Mega Kick
~ Foresight
~ Meditate / Brick Break
Hi Jump Kick is powerful and gets STAB, but has this nasty side effect that it may damage your own Pokémon if the move misses. Jolly boosts speed so that Hitmonlee may outspeed most of the Pokémon you'll encounter. Mega Kick for coverage, Foresight so that you may hit Ghost type Pokémon. Meditate boosts your attack, whilst Brick Break allows you to have a STAB without risking recoil.
