Assuming you're the captain, you should get a list of ~5-10 heroes that each player on your team feels comfortable playing and you should personally rate each of those heroes in terms of how effective you'll feel they will be. This creates a sort of 'internal tier-list' that you can compare with traditional tier lists and find which heroes you're lacking. For example, if no one on your team can play Naga Siren, who is considered extremely high tier right now, you should almost definitely be banning her or in the next few weeks someone on your team should be learning the hero.
Watch professional matches and see how they go about their drafting. Is the meta favoring picking pushers first, carries first, supports first, teamfight heroes first? and etc. Most teams you'll face will follow the current meta and so you should be aware of it and ban/pick accordingly, though you don't necessarily have to follow it yourself.
From there, have some practice drafts against another team's captain and make sure you've covered all of your bases with regards to ban/picks both from the first-pick and second-pick side. If you have a strategy for your initial bans, there's usually no reason to deviate from it from match to match unless you know your opponent. In doing this you might find a hero you frequently want to pick that no one is comfortable playing- is it worth someone's time to learn that hero?
Then play organized scrims with your time. For Dota2, #dota2scrims.na and #dota2scrims.eu on QuakeNet on IRC is your best bet to find opponents, but I don't know what is currently being used for original DotA. #dotascrim might be active, or my best suggestion would be to check out GArena and specify that you're looking for teams of 5 to play against. Playing public games as a 5-man team will get you used to uncoordinated opponents and create bad habits, don't do this for any length of time. After your games, win/lose, go over the replays as a team and watch the games for mistakes. Identify bad decisions, figure out how to prevent them in the future, and move on. Arguments and lingering on mistakes is much like calling someone out and is unnecessary.
Good luck! =)
For actual gameplay related stuff:
Draft to win your lanes. A lot of "above average" teams are used to just pubstomping together and don't know how to handle playing from behind with a more lategame-oriented team. When you get these teams out of their comfort zone they fall apart.
Ban out snowbally heroes that control the early-midgame like TA, Night Stalker, Nature's Prophet, etc. and pick up Lich, Venomancer, QoP, etc. If you want a DPS carry at all pick one that's useful in lane also, such as DK, Morphling, etc. or just pick a midgame carry like Death Prophet, Leshrac, etc. Win all of your lanes and take outer towers with your lead and stronger earlygame heroes. Pick up Mek/Drum and force more teamfights to take Rosh/T2's. Don't give up silly kills, they just delay your ability to push and give the other team an opportunity to make up their deficit.
Keep multiple wards up at all times and make sure everyone on your team is constantly carrying TP's. Force the enemy to commit to its defenses and commit to its aggression on your towers, punishing them if they don't do so.
If you do lose, it will be because you're outplayed in your lanes and so it will be clear where you need to improve, unlike some games where you lose and aren't quite sure what you could have done differently. This is a long-run approach. You're probably not going to win the tournament unless you're actually a team that deserves it, but maybe you'll win the 3rd one down the line if your team stays together and improves well.