How can I stay strong enough to avoid being eliminated while at the same time avoid being too strong and making myself a target?
Hold a medium-to-small sized continent (like South America or Africa) and avoid aggression while you build up. Attempting to take Asia, Europe, or North America will raise a lot of red flags, and Australia's defensibility makes it suspicious too.
RISK is defined in the later game by the ridiculous armies you get when trading in cards, so your main task is to be in the position to trade in for a lot, then bust out of your home and take another continent.
Don't underestimate the importance of visual cues or psychological warfare; in games like Risk and Diplomacy, these are the very underpinnings of your negotiation platform. For example, if you are playing one of the game sets that features little men and cannons, then arrange them on your spaces facing away from one enemy, and massed on the 'border' with another enemy. It has no in-game effect whatsoever, but sends a very definite visual message, which is interpreted on a subconscious level. When it comes to your turn to attack, mull over your options, or pretend to be doing some dice-math in your head, while staring down intently at a given enemy. Even if you don't actually attack, you've conveyed to the rest of the table that you WOULD attack that enemy, if only you COULD, which leads to the "He must be doing too well" syndrome from everyone else.
There are two sides to Risk, one of them being the actual armies, countries and cards you hold. As lilserf correctly points out, you usually aren't allowed to hold any major continents. I don't quite agree with Australia. It's easy to defend, but it's usually not near the tension and there's the danger that you get isolated, so I don't find it superior over South America or Africa.
Now the other side of risk that you shouldn't underestimate is your mouth. Making deals, persuading etc, can have a huge impact on how the other players react. The most dangerous thing is often a player getting a lot of armies without really having a plan of what to do. Here the weapon is not what's on the board, but what's in your mouth.
I enjoyed reading the answers to this quintessential question that defines the game of Risk. I find Risk to be a game of balance of power & playing your position is important. What you do when you are the dominating player is different than when you are the mid-player or the week player. When playing on line in real time you must also consider who are the reliable players in relation to connectivity as to who will drop next. In addition to the other answers posted here I would add the isolating method. If your opponent controls Australia or South America early on, you choose as position on the map far from theses continents & eliminate all soldiers of that player that are near yours thus forcing the incremental strength of that player to attack your opponents as opposed to you.