Historically, Intelligence was the first and most powerfully predictive variable of individual differences.
Personality came later - and was formalized by the work of H.J Eysenck as a separate explanatory variable which can be seen after Intelligence is controlled-for.
Personality evaluations are not 'tests' in the way that IQ is measured by tests. Usually Personality is evaluated from self-rating questionnaires - sometimes from ratings by other people (e.g. teachers or parents). Sometimes Personality is measured indirectly in terms of behaviours - for example Conscientiousness might be measured by how many hours a person spends studying, or by their record of employment attendance.
What is the formal relationship between Intelligence and Personality?
The answer is that Intelligence is hierarchically superior to Personality.
Because intelligence very obviously affects personality - but personality does not affect intelligence.
Intelligence is primary as an explanation of behaviour - primary both historically, and formally, analytically.
Thus, Personality is Secondary to Intelligence.
How can this be understood? Intelligence is a measure of brain efficiency - something like speed of cognitive processing. therefore it affects pretty much all cognitive attributes in a quantitative manner.
Personality is disposition, related to the set-up and balance of attributes. Personality could be conceptualized as a different kind (or balance) of cognitive processing (not a different speed).
So that when exposed to the same stimulus - different Personalities (of the same Intelligence) will process the stimulus differently, to lead to different behavours.
But to establish a difference in Personality, the Intelligence must be controlled - because the same personalities exposed to the same stimuli will generate different behaviours if they have different intelligences - because differences in processing speed will suffice to produce different behaviours.
This has not been well understood - indeed I did not properly understand it until a few hours ago! Until then I thought of Intelligence and Personality as independent ways of predicting human behaviour. But I now see this is wrong - Intelligence is Primary and hierarchically above Personality - and only after Intelligence has been sufficiently controlled-for should Personality be evaluated.
In other words, as a matter of routine - when measuring Personality, one should also do an IQ test.
And before looking at the effect of personality on behaviour, one ought to remove the effect of intelligence (by stratified analysis, preferably - i.e. creating narrow strata of IQ and only looking a personality effects within these strata - or else by some kind of regression).
To put it differently, it is legitimate to measure IQ without Personality - but not legitimate to measure Personality without IQ.
What this means is that:
1. Two people (or groups) with the same IQ but different measured-Personalities traits (as evaluated using self-rating scales, or whatever), nonetheless have the same IQ.
2. Two people (or groups) with the same measured-Personality traits but different IQs, in reality have different Personalities.
However, in practice, much personality research is done on an already-intelligence-stratified sample - such as Psychology Students at Mudsville State University - in these situations the researcher can usually get-away-with missing out IQ testing and just evaluating Personality.
(However, this does not apply to the Big Five pseudo-trait of Openness - which is often so sensitive to IQ differences that it varies even within strata such as the same class of the same college. If Intelligence is controlled-for, then the effect of Openness disappears - because Openness is merely 'the personality type of intelligent people in Western-type societies' - but rather badly conceptualized.)
The best conceptualized Personality traits, which derive essentially from HJ Eysenck, are Introversion versus Extraversion and Neuroticism versus Emotional Stability - and these are relatively robust to IQ differences (especially in college populations which provide most of the subjects).
(However, it seems likely that high levels of trait Neuroticism are contaminated by pathology; and likewise extremes of both Extraversion and Introversion. in other words, some personality differences are the result of diseases in specific people.)
The other major Eysenck trait is Psychoticism - which the Big Five splits into the inverse of Conscientiousness, and Agreeableness (which is the same entity as Simon Baron Cohen's Empathizing) - to which could be added Schizotypy to capture the 'psychotic' element of Psychoticism.
Ultimately, Personality needs to be related to some underlying biological, and adaptive, mechanism - and the most likely of these underlying explanatory models seems to be Life History; which is another topic for another time.