Of course it can be done.
It is easiest if the symbiote is unable to act on its own (outside of jumping to another host as you described).
The way you will do this is simple:
- Enhanced Trait 30 (Enhanced Strength 12, Enhanced Stamina 12,
Enhanced Fighting 6)
- AE: Flight Flight 6, Enhanced Trait 18 (Enhanced Feats(Move-By Action, Evasion 2, Favored Environment (flying)) Enhanced Strength 8,
Enhanced Strength 6 (Limited: lifting only))
- AE: Protective Barrier Protection 10 (Impervious 5, Shapeable) Enhanced Trait 5 (Ultimate Effort: Toughness, Ultimate Effort:
Reflex, Interpose, Set-up 2)
- AE: Symbiote Possession Mind Control 10 (Ranged Affliction resisted by Will, Extras: Duration: Sustained (+2/rank), Progressive(+2/rank), Flaws: Instant Recovery(-1/rank), Limited to 2nd/3rd degree(-1/rank), Limited to one target at a time (-1/rank)) -/Defenseless/Controlled
The total cost of this power is 30 point +1 point per alternate effect. Essentially, once per round you can shift your array around (the symbiote moves to a different area, or does a different thing). If you need, you can throw your symbiote as a ranged attack. If it hits your target makes DC 20 Will saves until you recall the symbiote or it becomes Controlled (see Affliction, page 87 of DC Adventures/M&M 3E). The symbiote will continue trying to take over your target each round, until it takes complete control (Progressive + sustained duration) or the target succeeds at a Will save, which completely forces the symbiote out.
You can buy any number of different powers for the array at one point each, and you have 30 points to spend on each power you buy for the array. If you put more points into the array (such as making it 40 points + 1/effect) you can enhance the number of points you have for each power.
Most of what the symbiote will give you is Enhanced Trait (which basically lets you buy anything you could at character creation) to boost your abilities or give you feats.
If your symbiote gives you some sort of continual effect while it's joined with you, just add that to each power in the array except the Affliction.
If your symbiote can exit your body for a limited time and act independently, you can add another Alternate Effect: Summon. Spend however many points your array has to build your Summon power, give it the Active and Heroic extras (lets it act independently and makes it not subject to the minion rules). At 5 points/rank, you could get a rank 6 symbiote in the above array, who you would build before the game using 90 power points (15 x rank).
If summon doesn't seem right (by the rules, the symbiote would simply disappear when incapacitated) you could consider having Enhanced Trait: Sidekick instead. Put 30 ranks into it (as per the power above) and you could build the symbiote as a 150-point character (assuming your character has > 150 power points, sidekicks have to have fewer than you). Or, you could do something like Enhanced Stamina 10, Enhanced Trait (Sidekick) 20 to have a 100-point Symbiote sidekick that leaves you with just enough of itself to make you tougher (or similar).
Side note: Built all the powers you want in your array, then make the most expensive one the primary effect. Primary effect doesn't mean 'what I'm using by default'. If you decide to go for a 50 point Summon, that should be your primary effect. Everything else would then be Alternate Effects built with 50 points. You don't have to spend all the power points available on each effect.
Edit: This assumes you mean M&M 3rd Edition (aka DC Adventures). If this is for 2nd Edition, there's a few differences, largely around the names of powers/feats, but the idea is the same.