I'm a guitarist but there's obviously a lot of cross-over regarding technique.
Don't sit down and try and play anything fast/wide stretches straight away, work your way up to it, look up some warm up exercises. Generally running up and down scales with various permutations for 5-10 minutes is better than nothing.
It sounds like you've identified that tensing up is not what you want to happen, people tense up when their technique isn't correct. You should search for some scales and arpeggios to play, get a metronome (or use metronomeonline.com), start to play them slowly, your aim is to be able to play them/whatever difficult bass parts you can already play at the same speed you can play them now but without tensing up. It will take a long time as you're effectively re-learning your technique, just start slow, increase the speed as you improve, keep relaxed, if at any point you notice you are tense, relax again.
By the way all of the above could be absolutely useless to you if there is a fundamental flaw in your technique somewhere, this could be down to any number of things, posture, left/right hand technique (which covers a multitude of things). It is areas like this where sitting down with a good instructor can help. I'd suggest going for a couple of lessons really, it'll cost money, but it is the right way to do things. Good luck.
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